your questions regarding the game rules (improving the interpretation).

Does the player need to count the cells?

Players already count movement points by cells, don't they?

Is a diagonal move considered the same as a horizontal move?

Diagonal move: 141, horizontal move: 100. There is a 50% penalty for sand. This is a well-known fact.

And if the boundary between the respawn center is blurred graphically, and it is not possible to accurately determine the type of cells for all of them? This happens on many landscapes.

It is interpreted in favor of the player; the essence of the rule is to avoid abuse of "illegal scouting."

And if some cells are covered by objects, and it is difficult to determine the type of territory under them?
Are mountains considered territory for the General or not? In general, many clarifying questions will arise...

Territories covered by objects are interpreted in favor of the player; mountains are not considered territory for the General. The General's territory is an area on the sand.

This is a rough version. If it is refined to eliminate ambiguities and add clarifying explanations with a screenshot and explanation of actions, it could be viable.