By American way do you mean cutting the food then transferring the fork to your right hand for eating? Or is there some other distinction?
This explains the difference. The European method seems the most optimal.
On the other hand, I don't think Americans ever pick up food with their fork and switch the loaded fork to the other hand, especially if the food is scooped, not speared. A lot of food would be dropped in the process.
As a non-conformist, I taught myself to use my knife in the non-dominant hand so that the fork is used in the same hand regardless of knife usage.
Or, it could be what my English son-in-law does, he uses his fork and knife, in different hands to aid in pushing food onto his fork. (He's right handed, not that it matters in this case.)