We did some analysis of Event Cart (April 2019) and advice was that it was going to be a lot of work, so we instead invested more effort in making Webform_civicrm work for multiple folk signing up for multiple events
A more custom approach could involve adding a Custom Field to Events, that is used to differentiate if the Event is 'public' or not, and then build an extension that provides the ACL to determine whether anonymous should be able to see/register the event.
The title of your question only makes a distinction for "anonymous" users; the question itself mentions "Member-only" vs "public".
If by "Member" you mean "current membership holders as defined in CiviMember", then the Member Only Event extension is probably a good choice.
If on the other hand you only mean "members of my site" a.k.a. "logged in users", ...
For word replacements you often have to use the english word as the original word, so you would put:
Events => Missions et PAD
For menu entries you can also edit it at Administer - Customize Data and Screens - Navigation menu. Right-click on Events and edit it to change the title.
I encountered this bug myself last week and fixed it; I forgot about this post until @herb posted a comment on it.
The fix will be in CiviCRM 5.25+, but the bug is logged as https://lab.civicrm.org/dev/financial/issues/119, and the fix is incredibly simple to apply.