I am looking for a chess engine designed with variant play in mind. Preferably it would be possible to modify piece movement to make computerized versions of new chess variants.
Having AI would be nice but is not essential.
Tabletop simulator is your go-to for that. Essentially it's a board game simulator that allows you to design your own boards and pieces and cards and whatever. There is no AI and you play by manually moving pieces according to the rules of whatever game your playing. It even gives you the option to screw the rules and just throw things around. There is support for online play and the feature list is very robust, allowing you to quickly shuffle cards as well as hide things from your opponent.
With your example, you can load in a chess board with the pieces, and you can then decide to move the pieces according to the new rules you made up. It does confuse me that you need a program for that. You can technically do that with a real life chess board and pieces. Tabeltop simulator is mainly meant to be a universal solution to the problem of playing board games online.
I am also looking for turn based strategy games where movement options provides extra depth. An example of this would be a tactical rpg where each unit has defined movement patterns.
"Into the Breach" might interest you then. You command a squad of mechs who are trying to defend a city from kaiju like aliens called the vek. Each turn, you will be informed of what the Vek are planning to do and you can do all sorts of things to disrupt their plan. You can damage and kill them of course, but you can also knock them around so they attack the wrong target (including other Vek), afflict them with CC so they can't do anything, or even have them run into environmental hazards.