Korach, Bamidbar 17:3. When People Who Sin Become Sinners את מחתות החטאים האלה.

The Gemara (Brachos 10a):

הנהו בריוני דהוו בשבבותיה דר"מ והוו קא מצערו ליה טובא הוה קא בעי ר' מאיר רחמי עלויהו כי היכי דלימותו אמרה לי' ברוריא דביתהו מאי דעתך משום דכתיב (תהלים קד, לה) יתמו חטאים מי כתיב חוטאים חטאים כתיב ועוד שפיל לסיפיה דקרא ורשעים עוד אינם כיון דיתמו חטאים ורשעים עוד אינם אלא בעי רחמי עלויהו דלהדרו בתשובה ורשעים עוד אינם בעא רחמי עלויהו והדרו בתשובה:

Reb Meir was plagued by wicked neighbors, and he intended to pray that they be dispatched to their divine judgment.  Bruria, his wife, said to him, it says in the passuk "may sins be eliminated from the Earth," it doesn't say "sinners."  Better pray that they repent and abandon their sins.  Reb Meir prayed for them, and as a result of his prayers they did do teshuva.  (The Maharsha there, and in Sukka on the same page, asks the obvious question, "how can one person's prayers influence another's piety?  הכל בידי שמים חוץ מיראת שמים, all is in controlled by heaven except for piety and fear of Hashem!"  Among the answers are: Reb Meir asked that Hashem ensure they meet righteous people whose example would inspire them; or that Tefilla can even help bring inspiration to other people, as every mother that prays that her child be a tzadik knows.  Even fathers say in birkas habanim שיתן בלבך אהבתו ויראתו ותהיה יראת השם על פניך כל ימיך שלא תחטא.  See also an excellent discussion of the Mahrsha at Chaim Brown's site.) 

Everyone there asks that Beruria's point seems incorrect.  Tanach is filled with examples of sinners- Chot'im- being referred to as sins- Chata'im.  One good example is our parsha, where Aharon was commanded to gather the incense plates of the supporters of Korach, who were killed by divine retribution for their rebellion, and they are referred to as Chata'tim.  את מחתות החטאים האלה בנפשותם.  

Additional examples from all over:

    תהלים נא, טו    אלמדה פושעים דרכיך, וחטאים אליך ישובו  

תהלים א, א      אשרי האיש אשר לא הלך בעצת רשעים ובדרך חטאים לא עמד ובמושב לצים לא ישב

בראשית יג, יג   ואנשי סדום רעים וחטאים לה' מאד

ישעיה א, כח     ושבר פושעים וחטאים יחדיו ישובו 

We even find this usage in Yiddish.  We refer to an unusually wealthy man not as an עשיר-wealthy, but as an עושר- wealth.

The Nesivos in his Nachlas Yaakov (and Emes L'Yaakov and Iyei Hayam) says two answers to this question.  

Answer I 

He brings from the Ramban in Breishis 25:28  יכנה האיש במעשהו לתדירותו, that a person that embodies a certain behavior can be given a name that refers to that behavior.  A person that is totally immersed in prayer can be called Prayer.  A person that is thoroughly and obdurately sinful can be called Sins.  (In English, the equivalent would be the difference between "A man who sins" and "A sinner," the change from adjectival phrase to noun.  But a better word would be sinnerman, as in the traditional spiritual song.)  This only applies when the person is so sinful that he cannot be defined as separate from his sins, they are a part of what defines him.  Divine assistance to do teshuva is withheld from such people (see, e.g., Rambam 6 Teshuva 3.)  Bruria was saying that only such people are called חטאים.  People who sin, but are not hopelessly sinful, are not called חטאים, they are called חוטאים.

שהחטא כבר נעשה עצמיות מהם ואי אפשר להתפרד מהם והן אותן שמונעין מהן דרכי התשובה והן ראויין לקללן משא"כ אותן שמכונין בשם חוטאים שהחטא אינו בהן עצמיות רק במקרה שאפשר להם עדיין בתשובה אסור לקללן דאמור ישובו עדיין

The distinction he makes in this answer is borne out with a simple glance at the unpronounced and therefore disregarded dikduk.  In the Mandelkorn, there are separate lists for חטא with a Patach and חטא with a Chataf Patach.  A Chataf Patach replaces a Tzeirei, as in Cheit, and חטאים with the Chataf Patach means sins.  If you're referring to people who have done sins, then you would say חוטאיםf. But when it says חטאים with a simple Patach, it also means sinners, but it's a different kind of sinner than חוטאים.  The difference is like the difference between גונב and גנב, between נוגח and נגח.  A Goneiv is a man who stole.  A Ganav is a professional crook.  When it says גנב or נגח or חטאים with a simple Patach, it becomes a label, a behavior that has become definitive of the person or thing.  So Beruria was asking Reb Meir, "Does it say Chot'im, which means people who sin?  No, it says Chata'im without a Chataf Patach, and that means a lifer, a recidivist, somebody who embodies sin."

All the pesukim we cited above have a regular Patach.

אֵת מַחְתּוֹת הַחַטָּאִים הָאֵלֶּה, יִתַּמּוּ חַטָּאִים מִן הָאָרֶץ. וְחַטָּאִים אֵלֶיךָ יָשׁוּבוּ, וְאַנְשֵׁי סְדֹם רָעִים וְחַטָּאִים,  וְשֶׁבֶר פֹּשְׁעִים וְחַטָּאִים יַחְדָּו וְעֹזְבֵי ה' יִכְלוּ

There are many appearances of חטא in Tanach with a Chataf Patach.  The only appearance of חטאים in that form is in Koheles 10:4.

אִם רוּחַ הַמּוֹשֵׁל תַּעֲלֶה עָלֶיךָ מְקוֹמְךָ אַלתַּנַּח  כִּי מַרְפֵּא יַנִּיחַ חֲטָאִים גְּדוֹלִים.

Bruria told Reb Meir that certainly, when a person is beyond redemption, one may curse him.  "יתמו חטאים מן הארץ"!  As I tell my shiur, Rav Huna says (Kiddushin 20a, אמר רב הונא כיון שעבר אדם עבירה ושנה בה הותרה לו הותרה לו סלקא דעתך אלא נעשית לו כהיתר,) that if a person does a sin twice, it becomes like it's muttar, he doesn't even think twice about doing it.  But what if a person does a sin three times?  What does he feel at that point?  I say that at that point he decides it's a mitzva.  For such a person, the possibility of Teshuva is vanishingly small.  But where neighborhood troublemakers are just Biryonim, there is hope for them.  I am sure that Reb Meir had rebuked them- there is a chiyuv de'oraysa of  לא תשנא את אחיך בלבבך הוכח תוכיח את עמיתך ולא -תישא עליו חטא (Vayikra 19:17).  But what Reb Meir had not tried was Tefilla.  Bruria told him "Don't write them off; True, they ignored your tochacha, and it seems that there's no hope for them, but first, daven for them, and let's see what happens."

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Answer II

The Nesivos says another teretz, and I think it's even more interesting than the first teretz.  

He says that for some people, avoiding sin is extremely difficult because of how they were created, whether it is a stronger yetzer hara or many other possible reasons.  From the Akeida in Shaar 63:

כי רבים מהם היותם על השיעור ההוא מהרוע לא היה רוע בחירתם אלא מצד חוזק חומרם ורוע תכונתם הטבעית שעלה כך בגורלם מהסבות שנזדמנו להם בעת הורתם אשר לא עצרו כח לכבשו. וכמו שכתב הרב המורה פרק ל״ג חלק שלישי כי מי שטבעו חם ביותר אי אפשר שיהיה ירא שמים בתאוותיו וכבר חשבו קצת החכמים שאין האדם רצוניי על הרעות וכל שכן אם הם נמשכות אל רוע תכונתם. עם שכבר דחה אותם הפילוסוף וביאר שהכל בידי אדם כמו שכתבנו בשער כ״ב מכל מקום דינא רבא ודינא זוטא איכא בינייהו כי אחר שהם לא בחרו לחם אלו ההתחלות הפחותות והטבע העלה אותם בגורלם וידו חלקתם לא בקו עד שאם יכבשום היה ראויין לשכר גדול כמו שאמר (אבות פ״ה) לפום צערא אגרא הנה כשלא עמדו כנגדם ראוי להמציא להם תקון 

So Bruria was saying that there are two kinds of sinners.  There are sinners that are born with strong evil inclinations, or amid terrible circumstances, and there are sinners that choose to do their sins out of a desire to be wicked.  For the people who purely out of wickedness choose to sin, prayer is a waste of time.  But for those poor people that were born with the bad luck of some inherent or circumstantial predisposition to sin, prayer is perfectly appropriate, because you are asking Hashem to level the playing field, to remove their unfortunate predisposition.  So Beruria told Reb Meir, when it says in Tehillim that David Hamelech cursed חטאים, he couldn't possibly be cursing people whose hot nature drives them to sin.  So he must have been saying יתמו חטאים, may Hashem help them by cooling off their burning Yetzer Hara.  Perhaps these people are of the nebach resha'im type, and for them, the proper tefilla is that they be freed from the terrible Yetzer Hara that drives them to sin.  As the Nesivos says, along the same lines, in his Nachlas Yaakov,

וזה ענין ההבדל בין חוטאים לחטאים כי חוטאים מורה על שהוא פועל החטא כי חוטא

על משקל פועל וחטאים מורה על שרשו שהוא תכונה רעה בנפש כאלו הוא נעדר הפועל ולזה אמרו מ׳ כתיב חוטאים חטאי׳ כתיב וזה אין שייך לומר שיבקש רחמים על אותן שימותו וע״כ הבקשה כמו שפרש״י שיכלה היצה״ר מהם ושיתרפא התכונה רעה מהם וזה אמרה דביתהו כיון שלא קילל דור רק אותן החוטאים ברוע בחירתם משא״כ אותן החוטאים מצד רוע חומרם אדרבא אבקש רחמי׳ עליהם שיתרפאו מהם א״כ אין לבקש רחמי׳ על אנשי׳ פ טים שימותו דדלמא חוטאים מצד רוע חומרם ולו נתכנו עלילות:

(Again, see the link to the Divrei Chaim I have above for a discussion of how Tefilla helps to restore Bechira.)

One of the reasons I like the second answer better is that the first answer doesn't work well in the language of the Gemara.  According to that pshat, Beruria was saying that the passuk in Tehillim was indeed a curse against recidivist sinners, but there people were not like that.  The problem is that she said that this explains the passuk ועוד שפיל לסיפיה דקרא ורשעים עוד אינם כיון דיתמו חטאים ורשעים עוד אינם אלא בעי רחמי עלויהו דלהדרו בתשובה ורשעים עוד אינם , "look at the end of the passuk!  There will no longer be wicked people- because Chata'im came to an end, the result is there will no longer be wicked people.  Pray for them that they do Teshuva."  According to the first answer, the passuk is not talking about people for whom you pray, it's talking about people whom you curse.

So, to summarize:

• The word חטאים can mean people who have sinned, and it can mean veteran sinners.  Rav Meir's neighbors were people who sinned, the congregation of Korach were sinners.
• Also, the Akeida that not all sinners are equally culpable.  Some people are faced with physical or emotional or cultural drives that are far harder to resist than those that others face.  They're all judged for their sins, but you don't throw the book at the ones that are nebach resha'im.   As the Baal Ha'Akeidah puts it, דינא רבא ודינא זוטא איכא בינייהו.  Does this support Rav Dessler's  "נקודת הבחירה"?  I don't know.