If I played a game on Xbox, Will the game progress be saved if I buy the same game off the...

No. That game does not support cross saves.

Save data is separate from the game data. So if you buy the same game digitally, you'll have the same progress.

I ran into the same problem in the past. You will still keep the game downloaded, when you try to play it, a message will pop up asking you to insert the disk or to buy the game. If you buy the game then you can start playing it right away. Plus all
of your progress from the time you played it previously is saved.

I’m playing halo infinite and I’m having the same issue at the tower. My progress will save and I’ll reach saving checkpoints. But when I quit the game and eventually resumed the game within 1 minute of quitting and leaving the game, my progress did not save. The same thing happened last night. Why does this keep occurring and what needs to be done in order to save my progress?

Actually GTA IV doesn't matter where you buy it from. It need two activation. One of Rockstar Games Social Club activation. This is necessary for start the game. And GFWL/Xbox activation to save game progress and playing online. I could play with xbox
live online before buying this computer.