Are you using an add-on to manage the propagation? If so I guess that's the bit that isn't working. There was an add-on for Lightning which did this but I haven't used it for ages. As far as I recall it used XML data from Google, but I read that that is being discontinued: https://support.mozilla.org/en-US/questions/1097284 You could try uninstalling the add-on, closing and restarting Thunderbird and then installing it again to make sure you have the latest version and the xml link still works. Personally, some years ago I switched to using CalDAV instead -- that way I can manage my Google calendar directly in Lightning, and it works without any problems. Your post implies I think that you have more than one calendar showing in Lightning and you have to remember which one is which. Why not have them display in different colours? I have two Google calendars which I use at the same time in Lightning but I can see at a glance which one I'm using because the entries have different coloured backgrounds in the daily/weekly/monthly grid.

Thanks. Just tried using Caldav instead but it's the same: "Home" events aren't send to the google calendar.

Home presumably is a local calendar. That means it has got nothing to do with your Google Calendar. If you want to sync your Google Calendar using Lightning you'll need to set up your Google Calendar as a network calendar in Lightning. https://support.mozilla.org/en-US/kb/using-lightning-google-calendar

The article is a little bit outdated. Just skip over the section 'Getting an XML link to your calendar' in the article. Lightning will prompt you for your Google user ID and password, and you'll have to confirm Thunderbird access to your Google Calendar.