AFAIK there are three kind of items:

1. Sold and shipped by Amazon

2. Sold by third-party, fulfilled by Amazon

3. Sold and shipped by third-party

I never buy (3) and generally avoid (2) when there is a possibility of counterfeit, but otherwise I agree with you (2) is fine.

I think there's a kind of 1.5 as well - Sold by Amazon, supplied by god knows who (co-mingled stock of varying authenticity - could by Amazon, could be Chinese knock off), fulfilled by Amazon.

That's actually not 1.5, that's 1 that you just described. There is no option 1 that isn't co-mingled with inventory of unknown authenticity that (unless it happens to be an item that's never had inventory sent in by a 3rd party, but you have no way of knowing that).

Depending on the item and seller "sold and shipped by third party" may actually be better - no mingled inventory, might even be direct from manufacturer.