I was expecting that VPN’s (maybe not all) also use their own customers as relays, but you don’t mention them. Am I wrong?

It's definitely not all VPNs (typically the ones you pay for will buy their own hardware and internet connections in the places you can route your traffic through), but it is a common situation for the shadier (especially 'free') ones, and in that case it's usually not just a peer-to-peer thing, they make money by selling your connection to people who pay because they want a large pool of relatively reputable and disconnected IPs to send traffic through.

You're right, a lot of 'free VPN' software also uses your own internet as a residential proxy. It's often mentioned in the ToS but the users don't read that.

Honestly, I expect any of those free VPN services that require an app install to be doing precisely this. At this point, I just assume anything with "free" in it will cost me more than if I just paid for something instead.

That kind of VPN is considered a residential proxy.