The in-game save files in the download version get stored in a big blob of data in the folder that the game creates (C:\Users\[NAME]\AppData\Roaming\coc2electron) which persists regardless of what you do with any given installation of the game. This is why your saves transfer between versions, since they're not linked to the folder you install the game into.
The browser version stores saves in a big blob of data shared with a whole bunch of other stuff, located (at least for Firefox) in your Profile folder. These also persist regardless of version, but clearing your browser cache or running cleanup utilities may delete the saves.
The saves created via the Save to File option naturally go into whatever folder you tell them to go to.