I talked to a guy who already built a hexapod or two, and also watched some more videos on robot kinematics and I have some thoughts about moving the legs:

Most people out there use servo motors for robots with legs. I suppose it looks like a logical choice, because they are easy to use and available in a relativ wide choice of sizes and strengths and don't need much additional hardware to work.

But there seem to be problems with them, too.

For one, they can be quite jittery, as Aecert points out in one of his videos.

Then, they usually have a working radius of at least 60°, but especially in hexapods, only a fraction of that is used, which wastes potential and resolution. This can, of course, be mitigated with some kind of transmission, but few builds seem to do this.

And finally, they do not absorb shocks very well, which i learned from James Bruton in his latest video.

This made me think that it might be best to use brushless motors with a very high transmission ratio (maybe a cycloidal gear?) and encoders for the actual positions. This way I can have reliable movement and by using back-driveable mechanisms absorb shocks and loads way better.

I also might be able to put all the motors in the central body of the robot, which would make the legs potentially lighter and/or more stable,

But these are just some thought for tonight.