I'm curious which type of normal map works better? DirectX or OpenGL normal maps? Which one to use for cycles and eevee?
Only DirectX applications use DX normal maps. And DirectX run only under Windows mainly due to license. For everything else you need to use OpenGL. For sure is good always check documentation.
On image below you can find simple way how to convert DX to OpenGL (or OpenGL to DX)
Blender uses the OpenGL format. This image shows an optimized version of normal map conversion, because it doesn't convert between number and color, so it is faster with Cycles and EEVEE:
This second picture shows how you can recognize OpenGL normal maps just by looking at them. Notice, that on OpenGL normal map it looks as if there is light shining from top right, while on DirectX it's from bottom right direction.