How exactly do "start of turn" affects trigger
When the start of turn triggers happen at 1.2 every card currently in play which has a start of turn trigger tries to trigger. Since it's the runner's turn, their effects go first. The runner can choose the order of their own effects, so you can gain a credit from data folding, and then use that credit to pay to install something off The Supplier.

What you cannot do is install something off The Supplier / Personal Workshop and then have that newly installed card trigger a start of turn ability; it wasn't in play for the start of turn signal, and so has missed it's chance.

So you are correct, and the Data Folding wouldn't have given him any money for that turn.