The ARPAnet (the electronic system) was never an internet (or catenet). It predated internetworking, and it was one of the networks comprising the internet we are using now (specifically, it was 10.0.0.0/8) until its demise in 01990. As a social institution, though, you are correct: the internet we are using now started out as an experiment inside the ARPAnet project.
...what year are you from?
The long now foundation
> established in 01996 to foster long-term thinking.
Our work encourages imagination at the timescale of civilization
Disclaimer: I am not affiliated with the Long Now Foundation, but I endorse the suggestion to read about them.
I think you are making a distinction without a difference.