From my newbie understanding, isn't "I think" and "I'm called" which is "My name is", the same phrase?
.... to moushimasu -> ... I think Bill to moushimasu -> I'm called or my name is Bill
Or did I get it wrong?
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"と申します" (to moushimasu) is a more formal way to say "と言う" (to iu), which both mean "to say". As you say "name + to moushimasu" is one way to introduce yourself.
Keep in mind while this can translate to the English phrase "I'm called ~", it doesn't literally mean that. "と呼ばれる" (to yobareru) would be a closer literal match for that, though not commonly used for introducing yourself.
Another way to introduce yourself is "name + desu", where "desu" just means "is". But the "と申します" I think is the safest option.
There are several ways to say "I think", but one of the more common ones is "と思います" (to omoimasu), which does sound a little like "to moushimasu", which maybe is why you confused the two.