After you have been studying Japanese for even a short time, you will find that the language is full of homonyms (同音異義語). つき　has 11 homonyms listed in the standard dictionary 大辞林 . You mention only two of those homonyms.

yes.”つき”(tsuki) has some different meanings. One is ”突き”, as you mentioned. It means "thrusting". And another is ”月”. it means "moon" and also means "month" And we usually use ”ツキ” which means "luck". For examole, when you drew the first prize in the lottery, you may say ”ツイてる！（I'm lucky!)”. In any meanings, we pronounce it in the same way I would be happy if it could be your help.