I found a solution: instead of having one TYP file per customised map, I have built one common TYP file with all the styles I need (constraint is then to use each code no more than once of course) and then by grouping the gmapsupp_francia together with 3 customised maps and one unique TYP file, the resulting img file is correctly displayed in my Etrex Vista HCX.
About the names, I found one solution also by playing on the order I use to combine the various maps.
My first attempts was in sendmap20 to add the gmapsupp_francia first, and then my own maps. The map property submenu selector of which map tiles (not sure if ‘tile’ is the right word) are displayed then starts with a big list of tiles from the whole France, and only at the very end 1 tile per customised map, quite difficult to to find in this menu. But if I do the reverse: add my own maps first, then gmapsupp_francia, the list begins with my 3 maps that I can easily find in the list of tiles (with their customised name in clear).

In other words, my multiple map works. I now can start to optimise my styles…