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Just a guess, but TIC-80 projects all have the extension .tic, while the more common extensions are only .rom and the others you mentioned. The binary and web files are exports, while the .tic files are the games' "source code", more or less. So that should be the reason why it uses .tic as the extension :)
I would suggest you to visit https://tic80.com/play to find other TIC-80 games and their .tic files as well. That should work I think! Also, you can find my games there as well!