Alright, I did an attempt at fixing this by reverting the
InstallScope='perMachine' addition, but while testing the release of a 0.18.1.1 with this fix, I could not get the upgrade to work properly from either a clean installed 0.18.1 or a 0.18.0 that had already attempted the upgrade to 0.18.1. The only case it would fix would be a 0.18.0 or earlier installation that would try its first upgrade to 0.18.1.1.
Since I think the per-machine flag should should be there, I've added a message to the update popup that instructs the user to uninstall Tiled and re-install it manually.
I'm quite fed up with installers, having had numerous issues with NSIS and now with MSI, that I'm really thinking about alternatives again and maybe would go with plain zip file distribution, as long as there would be some auto-update mechanism.