is there a proper name for this joinery - a sort of supportive saddle

We have a round post and we wish to put a round log on top of it so we flatten a bit of the roundness off of the top log so it won't roll off (see first image below).  The problem is that now the bottom part of that log is encouraged to split off.  So we've been making a little support saddle in the post (second image).

Did we invent something, or is this a well known (and named) type of joinery?

With dimensional lumber, I've only heard it called a "halving joint", but that refers to any joint where half the wood is removed from each piece.  I've mostly seen them done with half the lumber removed from one side of each board, rather than top and bottom like you are doing, and with exactly half the lumber removed from each, rather than a smaller piece like you are doing.  There are lots of types of halving joints for all different configurations, but I'm not sure if what you are doing really falls into that category.

The top drawing in the top picture is called a rabbet or rabbet joint. The bottom pic is called a double rabbet, or could be referred to as a type of half-lap. These terms are generally used in cabinet making, but to me I think the same terms can be used to describe similar joinery on giant timber framing.





Images source: https://www.finepowertools.com/woodworking/wood-joint-types/

I suppose if we were pressed to make up a name for it, we could call it a "rabbet saddle"

That little notch I heard referred to as a shoulder so my vote is for shouldered half lap.

"Shouldered" is a longstanding timber framing term for that feature, so shouldered half-lap sounds just right.