Do Muslims believe Noah was muslim?

You have to differentiate between two things as Abbas Naderi has correctly mentioned.
One is the litral meaning of the word "muslim": that is "the one who has found peace" or "the one who has submitted his will/himself to the will of God". In this sense every prophet and every follower of a prophet at those very times have been "muslims". In fact the Qur'an has called Abraham "muslim" in the literal sense (here translated as "obedient"):
"Abraham was neither a Jew nor a Christian, but he was ever inclined to God and obedient to Him, and he was not of those who associate gods with God." (3:68)
As per this understanding every prophet of God has been a muslim.
And there is another concept one has to consider first. In a tradition (Hadith) of Sahi Muslim (one of the most authentic Hadith Books of Islam) it is said that the Holy Prophet of Islam, Muhammad (peace be upon him) said: "Every prophet before me had the duty to guide his people to good deeds, he knew, and to prevent them from doind evil deeds, of which he knew." And the Holy Qur'an tell says: And We did raise among every people a Messenger, preaching: ‘Worship Allah and shun the Evil One.' Then among them were some whom Allah guided and among them were some who became deserving of ruin. So travel through the earth, and see what was the end of those who treated the Prophets as liars! (16:36)
Alas, every prophet had more or less the same core message according to the Qur'an. Some varied in their messages due to circumstances of time and space. In this sense again every prophet and his followers have been "muslims" because the core message has always been the same, to be obedient to God or to submit your will to God.
The other meaning of "muslim" is to be the adherent of the religion of Islam, which again actually puts every Muslim into the test to become a "muslim" in the literal sense to obey to all the commands of Allah and to lose himself in (the will of) God.

Yes, Muslims believe Noah, peace be upon him, was a Muslim. Here is the evidence from the Qur'an. "Say (O Muhammad): We believe in Allah and that which is revealed unto us and that which was revealed unto Abraham and Ishmael and Isaac and Jacob and the tribes (of the Children of Israel), and that which was vouchsafed unto Moses and Jesus and the prophets from their Lord. We make no distinction between any of them, and unto Him we have surrendered" (3:84). What this text means is that Muslims believe in the revelation that Muhammad (S) brought and we believe in the revelations of other prophets. Note that Muslims believe in the original revelations which means we do not believe in the modern Bible because many religious leaders changed the text of the Bible over history. But prophethood is not something unique to Muhammad (S). It is a pattern that goes back to Adam, pbuh, and repeated over and over again.

Further, there are several places in the Qur'an that show that Noah, pbuh, was a prophet. Here is one: "Lo! Allah preferred Adam and Noah and the Family of Abraham and the Family of 'Imran (father of Moses and Aaron) above (all His) creatures" (3:33).

This is a linguistic inaccuracy, a lack of precision in language.

Muslim, has two meanings, one is the concept that God talks about in Quran, and by that meaning, all theists have been Muslims, including all prophets.

The other one is the term we use to call adherents of Islam today. By that standard, Noah and other prophets (except Muhammad) were not Muslims.

Yes. He was a Muslim, although he can have Islam with another name. I mean he was Muslim as believing in Islam, but perhaps with another name ... Since he was before Prophet Muhammad (peace be upon him and his family)

But to the best of my knowledge, totally, all of the prophets actually had the same religion, but with different topics ...

Yes they very much do, in fact, Prophet Noah is among major prophets in Islam. There is a complete Surah in his name in Qur'an.

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