I can think of "that is to say", "in other words", "put differently". And I'd like to know if there are any subtle differences in the usage of these synonyms. Can they always be used interchangeably regardless of the context?
“i.e.” always, only, and ever means “that is”, and should always be read that way. It never means “put differently”. It is never meant to be read aloud or in your brain as eye-eee initials; it should be expanded for the abbreviation it is, and very most preferably into English not Latin.
You may be thinking of videlicet < vidēre licet, abbreviated viz. in print but originally a scribal abbreviation employing Tironian et: vi⁊. This is usually read aloud as “namely”, and has a slightly different nuance compared with i.e., which you can look up in any general reference book.
I very strongly recommend not using Latin abbreviations, because I can guarantee you that between reader and writer, someone won’t know what these properly mean. After all, look at the question you asked.