R. Ya'akov Bachrach (Nimukei HaGRIV, printed in the back of many editions of the Talmud) has a note on this Rashi, referencing a similarly worded discussion in Chullin 104b:
מתני׳ העוף עולה עם הגבינה על השולחן ואינו נאכל דברי ב"ש וב"ה אומרים לא עולה ולא נאכל א"ר יוסי זו מקולי ב"ש ומחומרי ב"ה...גמ׳ רבי יוסי היינו ת"ק...הא קמשמע לן מאן תנא קמא רבי יוסי כל האומר דבר בשם אומרו מביא גאולה לעולם שנאמר (אסתר ב, כב) ותאמר אסתר למלך בשם מרדכי
MISHNA: The meat of birds may be placed with cheese on one table but may not be eaten together with it; this is the statement of Beit Shammai. And Beit Hillel say: It may neither be placed on one table nor be eaten with cheese. Rabbi Yosei said: This is one of the disputes involving leniencies of Beit Shammai and stringencies of Beit Hillel...
GEMARA: The Gemara challenges: The opinion of Rabbi Yosei is identical to that of the first tanna...
[T]his is what the mishna teaches us: Who is the first tanna? It is Rabbi Yosei. The identification is important, since whoever reports a statement in the name of the one who said it brings redemption to the world. As it is stated with respect to the incident of Bigthan and Teresh: “And Esther reported it to the king in the name of Mordecai” (Esther 2:22), and Mordecai was later rewarded for saving the king’s life, paving the way for the miraculous salvation.
(Translation and elucidation from sefaria.org)
Rashi there comments:
והתנא שכח ולא הזכיר שמו בתחילה וחזר והזכיר שמו
And the tanna forgot and did not mention his name at first, and then went back and mentioned his name.
Bringing this all back to the passage from Shabbat under discussion here, it appears that your first option is correct - Rashi means that R. Menachem is the author of the first part of the beraita.
To answer your questions:
a) Rashi knows this because of the seeming redundancy of R. Menachem's words. R. Menachem's statement that this is a stringency which applies to erasing over writing does not add anything to the first statement (just like R. Yosei's statement in the mishnah in Chullin does not add anything to the tanna kamma's words). Something else must be going on here, and we conclude that, indeed, the sole purpose of R. Menachem's concluding statement in the beraita is to point out the he is in fact the author of the entire beraita.
b) The apparent implication that the tanna kamma's words were not actually spoken by R. Menachem can be explained like Rashi says in Chullin. The first part of the beraita was erroneously taught anonymously. This was subsequently corrected by adding a seemingly superfluous addition in the name of R. Menachem to point out his authorship of the first part.
c) It is important for us to know this because of the principle that whoever reports a statement in the name of the one who said it brings redemption to the world (as well as explaining away the seeming redundancy of R. Menachem's statement at the end of the beraita).