Does NAT leave a trace of the internal source?

Solution 1:

That question has no good answer because you ask a very specific question - for a generic term.

NAT is an approach, not a program. Every router implementing NAT - may do so differently. Implementation detail.

So, there is no generic answer.

IIRC back in the day you could not only identify NAT was used, but also certain router producer (based on such implementation details) though that was YEARS ago - things have cleaned up then. Seriously, this requires analysis for every router and possibly differs even between firmware versions over the years.

Oh, and another one - your example is bad. REALLY bad:

Let’s say I do a series of DNS requests from my machine to a public server

In any network setup I hever have seen, you would not do that - your machine would do the DNS request to the router, which also acts as a local DNS server. Any request it can not fulfill would be forwarded. The reason for this is i.e. finding local assets also registered in the DNS... which the remote DNS would not know. Anyhow, using this (normal) setup, there would be no NAT involved as any outside DNS request would really come from the same device.