usually you should not need to copy/paste the remote-cache from one pc to the other one. It is stored in steam cloud and you get the files from the steam cloud automaticly. You can check Steam cloud content manually , e.g. here’s the link : Playtime gone missing [Packages out of date] - #31 by MichaMMA
Worth to try is the Disable-Steam-Sync trick , which @EnsiFerrum666 mentioned. But in case there are still files in steam cloud, steam should ask you after re-enabling which files you want to use.
PS:: shareing your steam userid is not best idea. remove that allways from links you post…