The site switcher allows five sites max when they're not customized.
These are sorted in order by most reputation first. If you don't have a profile on five different sites, it will only show the sites you do have profiles on.
In your case, you have several different sites with the same reputation (101). So, how do we choose those? They're alphabetical (and cached).
I tested this to show it's the case:
On a sock puppet account, I had a handful of profiles including Parenting, Computer Science Educators, Meta Stack Exchange, Android Enthusiasts, and Arqade. Those five were in my site switcher, in that order:
Looking at yours and mine I guessed that the remaining sites are in alphabetical order. This explains the order and why The Workplace didn't show up in your list when you added it... so I logged in to my sock and joined a site that's earlier in the alphabet than Android Enthusiasts - 3D Printing. I refreshed the page... and nothing happened. Oh, yeah, caching. So, I edited my list and "reset to default" and ta-da!
Now 3D printing is on the list instead of Arqade:
You can follow these steps to try it for yourself if you're interested.
Do note, if you edit your list, it's pretty flexible. You can make it as few as one site and ... I haven't tested the upper limit but I'm guessing it's high-ish. You can also add meta sites if you'd like to have direct links to them, though there's a bug in the reputation listed. Do note, if you customize the list, it will be locked in that order until you reset to the default, which will re-sort by reputation and return to five sites (or less if you have fewer).