Can anyone tell me, what is the difference between 住んでいたい and 住みたい?
Both seem to be valid words... But I'm unsure which I should be using.
住みたい means "want to live" and is the default choice.
住んでいたい is its progressive form and is used when there's some sense of progression, which works best when you're already living where you want to live "progressively".
One overlap of usage is, e.g.
This is not a perfect analogy, but in English we have roughly corresponding
- I want to live in Tokyo forever.
- I want to be/keep living in Tokyo forever.
The second sentence can really only be said by someone already living in Tokyo, with a strong attachment to Tokyo. The first would usually be used by someone, who wishes to move (and live) in Tokyo. When the first sentence is used by someone already living in Tokyo, then it might suggest a more neutral intention.
Just for reference, searching the Balanced Corpus of Contemporary Written Japanese, I find
住みたい 133 results
住んでいたい 5 results (without false positives)
住んでたい 0 results
Of the 5 examples with 住んでいたい, 3 were using ずっと or いつまででも and all were referring to people already living where they want to be/keep living.
Apart from the idea to "keep living where you're already living", which has been suggested in other answers, you can also use 住んでいたい as an ongoing progressive action at some point in the future.
E.g. if you live in Tokyo now, you can say
五年後は沖縄に住みたい (suggesting you want to move there in 5 years)
五年後は沖縄に住んでいたい (you might move tomorrow or in 4 and a half years, but 5 years from now you want to be living in Okinawa)