Halo Master Chief Collection progresses the same on Xbox and PC?

If I start the Master Chief Collection on the Xbox and have played the campaign for a few hours, can I continue playing on the PC at the same point? (With the same Microsoft Acc then on the PC)

In theory, yes.

Unfortunately, I can't tell you at the moment whether the game is not reset on the PC or not

Yes, everything is the same, at least with the PC MS and the Xbox version. I don't know about Steam but I guess it's the same there because you have to log in to the Steam version with a Microsoft account.