If the joint is mitered, then the joint will open up with seasonal changes (if it lives in a part of the world with seasonal moisture changes).
Check out a picture frame made of wide solid wood. The inside corner will open if gets dryer and the outside will open if it gets more humid.
There is a difference between a picture frame and an L-shaped bar top. The only way two mitered pieces of wood will open as you have indicated is if the ends of each piece are solidly held in place.
Let's say that whereever the original poster installed his bar top his maximum seasonal moisture content change would result in 5/16ths total width change in a piece of 20 1/2" wide walnut. So the total amount of change between the two would be 5/8".
Lets say teh OP installs the top at its maximum moisture content (highly likely).
So if the OP did not rigidly fasten the short leg of the bar top, and dog boned the miter, the miter would not crack at all, it would simply change it's angle. At its maximum shrinkage value we started with, that angle would be 88.226 degrees instead of the 90 degrees he started with.
With the additional length of the short leg added to the total miter shrinkage, that short leg would move .9911 inches total at the end of it, assuming the long leg was rigidly fastened. One inch. If he could deal with that, then it could be mitered without cracking and with the entire joint glued.
I did just that on a solid cherry L shaped desk in 1982 in Colorado, and it was fine even after being moved to Houlton Maine in the early 2000's. I used 4 or 5 dog bones and glued the joint. The base just moves back and forth with the movement of the top.
The OP went with the butt joint and cannot glue any but three inches anywhere he wants to in the joint or risk either splitting the top or breaking the glue joint. If he glues on either end of the joint he will get 5/16th shrinkage on the other end of the end grained board. If he glued in the middle, he'll get 5/32nds on each end of it. If he used no glue, he will get it in the direction of the dog bone with the least torque. Hopefully he made the channels wide enough for the connectors to rack as the butt end board moves.
I'd have splined the joint myself, to insure that both boards remain on the same plane, and see no advantage to a half-lap joint in this application. Hopefully, the OP will give us an update in a year.