No coins or notification: Do I need to do something to get earned coins?

Based on the comments, this could be a display bug where your Snorlax was in fact knocked out of the Gym after your other Pokemon that brought home the coins.

There is one way to find out exactly what happened: The journal. While it has a limited number of entries, there is a seperate limit for gym entries and it now displays the coins, too. If you check the journal from your profile page, there will be entries including date and time of all gym events, including "Snorlax fought hard and returned".

If there's an entry for something else at the same date and earlier time, it's a display bug and your other Pokemon simply wasn't shown yet (and the coin total in the Shop apparently only updates once the notification at the top of the screen appears). If Snorlax did in fact return earlier than the other Pokemon, there must be something else going on.

By the way, you can check this many days later still. Due to the seperate entry limit for gym events, the end of the journal has many of these, usually dating back several days, even weeks if you weren't very active taking gyms.