I think that the issue differs because it's when I try to play offline (certain networks won't allow to connect to Minecraft servers). Over a good network, I was able to sign-out and sign back in. Microsoft asked for me to agree to some new terms of services. I then turned my wifi off, relaunched the MCL to test it and I could again play offline.
I think it has something to do with accepting terms of serves by Microsoft that solved it. I will keep monitoring the issue to see if it happens again.
I suspect I know what is going on. The issue happened again.
I think that because the MCL couldn't validate my login status (due to wifi connection to a Minecraft-blocking-network) it set some flag to say I'm logged out (while I'm not).
Steps to reproduce:
Launch the MCL with PC wifi connected to a network that blocks all Minecraft domains
Turn OFF PC wifi
Close the MCL
Open it back up
You'll find yourself in Demo mode and cannot play offline