Agreed, on all counts. This will be fixed in the next build!
Edit: This has always been the behaviour, except for one recent build in which the ‘save’ button wouldn’t appear at all no matter what. You’re supposed to edit the save game name, before you can save. I think that was more intuitive using the old text editing widget, because it put an underline under the game name. The dialog layout probably needs some general love to get users’ eyes drawn to the right spot, again.
But additionally, as you point out, the button should never vanish; it should just become disabled when no save game name has been provided. And the behaviour should be the same on subsequent openings of the dialog. (Maybe that whole “no save game name has been provided yet” thing should go away; maybe we should just provide a default name which the user can use if they want?)