Here an image of an old IBM-PC: https://www.7dayshop.com/blog/wp-content/uploads/2015/02/Ibm_pc_5150-1024x951.jpg Note that The dark area here is for floppy Disks. One to boot from and one for data, so it probably doesn’t have a hard drive. I think it looks pretty much like the one in the meme. Also, computer desks like this were back then very common at home and probably not in offices or universities. Furthermore, he has several boxes with floppy disks beside the desktop. A dumb terminal would have no need for floppy disks, but one of the old PCs would need a lot of them.
So all in all, I think it is most likely a real computer.
Syncing the calendar db and Todo.txt file with syncthing should also be possible and would be a solution that doesn’t need a server at all.