Method 1/2/4 don't have the search nature mechanic. That's why without box Method 1 is the most common, it doesn't take much time to generate the pkm so less time for vblank to mess things up.
I have never seen Method 3 or H3.
Fishing falls under the H methods. The nibble chance is separate from generation.
As Sabresite mentioned the H methods are for wild encounters and non-H methods are for stationary encounters.