I think the main menu screen has the release number on it somewhere so you could check there whether the versions are the sam (alternatively you can press “Feedback” in the launcher to pop up a form that will auto-populate the version number). Saves are definitely not compatible across patches, so if patch 9 hasn’t been applied to one machine then that could explain the problem.
I know that it’s possible to keep playing on earlier versions of the game but have never tried this myself. But if you haven’t actively done this, then it’s odd if Steam isn’t finding any updates for it. Do you have it set to auto-update and have you tried manually checking for updates? I think there’s a “Betas” option in the Steam game properties that you use to continue playing older patches, so you could also check that that says “None” on the machine that seems to be running the older version.