I only learned Horizontal Finger spins in Jul-Aug so I’m not an expert, but this does give me a successful-beginner’s perspective to share.
There’s a YouTube Video that explains the non-centering dynamic you were asking about. Do a YouTube search for “horizontal finger spins” and it should be the #1 return. (runtime ~13:39)
What I’ve found (personally) is that the Yoyo used is a huge factor in the finger-spin.
Wide shallow cups like the Horizon (for me) have too much circular motion and never center on my finger. I can use them… but they’re much more difficult for me. It’s worth noting: This is an inexperience problem on my part… others can finger-spin very very well with these Yoyos. (The C3 Move is like this for me… It is one of the regaled champions of the finger-spin, but I cannot get it to be stable to save my life).
---- My point being… I’ve formed the opinion through experience, for a beginner, some Yoyos that are championed as great for finger-spins by our veterans, may actually be difficult to learn on.
My favorite Yoyo’s to fingerspin on have been the Skyva and Firrox. The shutter can be really good, but you have to catch it in the inner cup.
I have been guilty of purchasing more Yoyos than I need. My collection is larger and more diverse than my skill-level really warrants. (just > 1 year of throwing) However, it was awfully nice to have a diverse collection while I was learning horizontal finger spins, to deduce what works and what does not.
For string I use Poly-Contest string almost exclusively. It works well for me, but I haven’t tried enough of a variety to claim it’s better or worse than anything else.
Again, My personal favorite Yoyo’s to fingerspin on have been the Skyva and Firrox. Shutter is pretty darn good also if you can hit the center cup.
Just my .02