We also need to understand that we are discussing "as things are given the actual products available". The actual strength of magnets used in magnetic boards, the type of hinges used in hinged boards, etc. We are not speaking about the theoretical (hinged boards just as easy to move as slotted if those were locking hinges --- simple design there were enough go boards to just if making them -- but would have to turn the board over to unlock before folding it).

Several years ago I was in the same boat as the OP with an identical budget. $200 buys a lot. You might consider tracking down a (semi) local woodworker and having a custom board made. It's more footwork, but the result is worth it.

I have an identical board. It is indeed katsura. They used to sell them at Samarkand when it still operated as an online retail outfit. It is a nice board. Whoever won the auction will be quite pleased with it. Each of the two sections is made from two joined pieces of wood. The line is almost invisible.

Unfortunately, my board spent years in storage under less than optimal conditions. It is still a very nice board, but one of the panels on one of the sections has discolored at a different rate than the others. I should live to replace it, but I'm in the same situation as OP. It is really difficult to find that quality in slotted boards these days.