For anyone stuck on the "Press Any Key" screen, this is an issue with your remotecache.vdf file for Payday 2. If you only care about playing offline, you can do this:

1. Disable Steam cloud for Payday 2.
2. Navigate to your "Steam\userdata\SteamID\218620" pathway.
3. Delete the remotecache.vdf file.
4. Move your saves from the "remote" folder to somewhere safe just in-case you want to play modern Gayday 2 again.
5. Go offline on Steam.
6. Start the game, you should be able to get past the "Press Any Key" screen.

This generates a new remotecache.vdf that contains information about the offline/old build save files. Going back online on Steam and launching the game (And maybe even having Steam Cloud Sync enabled for the game without launching it) re-adds the information from the old remotecache.vdf you deleted since it is fetching that information from Steam's servers. This will cause you to get stuck at the "Press Any Key" screen again on the old builds.

If you want to go online on an old build, you need to delete/reset all your cloud information for Payday 2. Steam for no good reason doesn't have a delete button for your cloud saves. There is however a work around I learned from here[www.saintsrowmods.com] (Big props to FusionH2o for figuring this out) to actually completely reset your cloud data for a title. I'll use Payday 2 as the example here. I added in some steps from above so you can generate save files for your old Payday 2 build.

1. Start Steam.
2. Disable Steam Cloud for Payday 2.
3. Delete all files and folders in "Steam\userdata\SteamID\218620" (So the "remote" folder and the remotecache.vdf file. Make sure to move your saves from the "remote" folder to somewhere safe just in-case you want to play modern Gayday 2 again).
4. Go offline on Steam.
5. Start the game to create fresh saves files.
6. Exit the game and Steam.
7. Delete remotecache.vdf in "Steam\userdata\SteamID\218620".
8. Start Steam and go online.
9. Enable Steam Cloud for Payday 2 and "Sync conflict" should appear in your game status.
10. Start your game and the "Cloud Sync Conflict" dialog should appear. DO NOT select any options.
11. Delete all files and folders again in "Steam\userdata\SteamID\218620" (So the "remote" folder and the remotecache.vdf file).
12. Go back to the "Cloud Sync Conflict" dialog and click on the option to use your local files. Steam will sync your "218620" folder that we just deleted all the contents from, meaning that all the files on the Cloud will be deleted as they are replace with nothing.
13. The game should start automatically, so alt-tab the game and disable Steam Cloud.
14. Return to the game and exit it (or kill it through task manager).
15. Exit Steam.
16. Delete the "218620" folder.
17. Re-open Steam and profit. You should be able to re-enable Steam Cloud for Payday 2 as well at this point.

Hopefully this helps, big thanks to FusionH2o, and have fun!