I have some on a 2D plane, and I extruded some more vertices across the 2D plane so that these vertices are overlapping with the plane, but I want those vertices to become an inherent part of the plane as if I just used a knife cut on the plane, is there anything like that?
Show us what you mean so we don’t have to guess
Tools available to you - knife cut (snap to vertices) or knife project, depending on the effect you want to achieve
https://docs.blender.org/manual/en/dev/modeling/meshes/editing/subdividing/knife_subdivide.html
You shouldn’t have the guess at anything, it’s a very specific circumstance so if you were experienced with the situation you’d be able to answer it. It’s okay that you’re not some omniscient god, alright? You don’t have to pretend anything. The answer is that blender currently has no known method for accomplishing this. The only option would be to work with individual edges at a time, press the space bar and look for “cursor to edge intersection” then press “shift+A” and add a single vertex and build the faces from scratch.
Or, separate the faces you want to merge and use the boolean modifier. No idea why blender wouldn’t have an edit mode version of the boolean modifier, but then again very little of blender is every intuitive.