I have the full sc2 and I'm playing with someone who's using the starter edition.
I know the starter edition can't use "Find match" in "Play cooperative" to find players via matchmaking.
If I create a custom game (I chose Discord IV -- one of the three starter edition four player co-op maps) and set category to "Co-op VS AI" and mode to "Medium" and invite the starter-edition-user then it all looks OK. The AI level seems indeed to be "Medium". After we win, neither of us gets a credit to the co-op achievements.
The starter-edition-user can't create the game themselves because the hardest mode available is then "Easy".
Is there something I'm doing wrong, or is this intentional or (better) documented?
I can't rule out the possibility that you can only get these achievements via matchmaking (I've always done it that way in the past), but I've seen videos of people seeming to get them with their friends (rather than random people), so I assume it is a starter edition thing.
EDIT: As stated by F.S. below, you have to use "play cooperative". People playing with their friends must be using "create party", then matchmaking via "find match" will say "looking for players" for a second, but will just use your party (this can be seen at 3 seconds in this video).