Shem keli is a special halachical status of utensil, following the first pshat of the Rash, this status is lost when the Din of utensil is irreversibly cancelled. The statements regards the piece of the former utensil, not the former utensil itself.
Now we need to resolve the contradiction.
Rash Keim 3.3, (to understand it we need to read the Rash in 2.2. before)
כשניקבה. במוציא זית וטהרה כשחזר ועשאה בזפת מקבלת טומאה מכאן ולהבא: חזרה ונשברה לשברים הרבה ואותו שבר שבו הזפת מחזיק רביעית טמא
An amphora was perforated and cannot more hold olives (the opinion of Rabbi Meir), it became clean, it is no more an utensil. He clogged it with tar. The amphora is back an utensil, because it has a possibility to become back a keli, we say that the Shem Keli (name of utensil) is not cancelled. Further, the Amphora was broken a second time, one of the pieces is able to hold Reviit. This piece is able to be unclean, this is a new utensil. The original amphora was a big a large utensil in terracotta. In Chapter 2 mishna 2, they taught that a large utensil (between Log, i.e. 4 X reviit, and Sea, 96 X reviit), when it's broken and we have pieces, a piece that can stand up and hold 1/96 of the original amphora volume becomes itself a new utensil. When such a piece was perforated it's no more an utensil, even if we repaired it with tar. So, when a piece-utensil is perforated it loses definitely the name of utensil. The source is the Tosefta (0)
There are two issues,
The original amphora, when is it no more a an utensil may reach its status again, (It is not an utensil but has still a name of utensil). For an utensil, losing the status of utensil is reversible, we say that the name of utensil is not cancelled. E.g. if we clog the hole by tar, the status of utensil comes back; (1)
and the pieces, when they did lose a status of utensil may not reach their status a second time. (we say that the name of utensil is lost). (2)
Let's imagine we have two identical pieces of terracotta they can hold water because they are clogged by tar. The first is an utensil and the second is not an utensil. The Chiddush in Mishna 3.3 is that we have a piece, that is an utensil thanks to the tar. The reason is that it's first step as utensil was thanks to the tar. The tar was there before the large amphora was broken, before the new piece-utensil was created. But if a piece has a status of utensil and loses it, it cannot come back. For a piece, losing status of keli is definitive, we say that it is losing the name of utensil.
The expression ביטול שם כלי in the Mishna refers to an irreversible change.
About what the Mishna says that the name of utensil was not cancelled?
The statement "the name of utensil was not cancelled from it" refers to the piece ("shever" is a masculine name in Hebrew), and therefore, the Rash says that the spelling would be better if it was:
טמא מפני שלא בטל שם כלי מעליו.
טמאה מפני שלא בטל שם כלי מעליה.
The Mishna speaks about the piece which is an utensil thanks to the tar. This is possible because the tar was here before the piece becomes for the first time an utensil. I hope that now it clear. The name of utensil was not cancelled from the piece before it was never brocken.
The comment Mishna Rishona gives several explanations.
When we see an amphora, even if it is perforated, despite the hole people call it "amphora".
When we have a perforated amphora we don't not give up using it, we say it must be repaired.
In name of the second pshat of the Rash: When an amphora was perforated by a large hole and cannot hold olives, we can decide to use it for larger fruits as pomegranates (we can without changing nothing, define a new use, and instantaneously it is back an utensile). But The Mishna Rishona has objections and says that the Rambam disagrees with this explanation.
When the Rash says:
ועוד דעדיין כלי היא לקבל בה רמונים וראויה לגרוגרות ולאגוזים
this is a second pshat, hard to get (see comment Mishna Acharona). Before this ועוד, the בטל שם כלי meant "it doesn't lose it's ability to become back a keli. In my answer I didn't adress this Pshat.
(0): Tosefta 2.3
חרש שפירש מן חבית ומן הקדרה אם מקבל כשיעור טמא אם לאו טהור נקב ועשאו בזפת טהור שכלי חרש שטהור שעה אחת אין לו טומאה לעולם
A piece or terracotta that comes from an amphora or from a pot. If it can hold a certain volume (1/96 of the first utensil as explained above) it is unclean (able to be unclean). If it cannot hotd this volume, it is clean (not able to be unclean). (if it can hold the volume) and afterward was perforated, and further clogged with tar, it is clean (not able to become unclean) because utensil of terracotta that became clean at one time will never become unclean.
(1): Rash mishna 3.3.
שאני הך סתימה דנעשית קודם שבטל שם כלי ממנה: וחבית שניקבה שם חבית עליה אע"פ שטהרה
(2) Rash mishna 2.2:
ויש לחלק בין כלי שלם אלא שניקבה לשבר כלי אלא שעשאו בזפת מפני שבטל [שם] כלי מעליו
אבל חרס הפורש מן החבית בטל שם כלי מעליו אע"פ שלא ניקב לפיכך אין מועלת סתימה לנקב שלו ובדין הוא דהוה ליה למיתני טמא לשון זכר דאחרס קאי ולא אחבית