https://jbiol.biomedcentral.com/articles/10.1186/jbiol105/fi...
It's fairly reminiscent of how limb growth looks during embryogenesis - and I think considering that is how other vertebrate (such as axolotl) regeneration looks, that it's probably what human limb regrowth would be like too. So your second description sounds more like it, but with continuing development of 'prior' parts while the 'later' parts (fingers etc) are still developing.
Michael Levin's group have experimented with a 'bioreactor' that can be attached to the stump of an amputated limb in frogs (that don't ordinarily regenerate limbs) and after a short period of treatment (days), the reactor is removed and the limb then regenerates on its own over the next few months. They are looking into reproducing this in rats now, with the clearly-stated aim of similar human regeneration being the ultimate goal.
I wholeheartedly recommend watching some of Levin's lectures on YouTube, there are a great many of them and I think they're both illuminating and fascinating!