DARPA was originally ARPA. They were under the department of defense but their project scope wasn't limited to defense projects. The reorganization that rebranded the agency as DARPA and made it defense focused ostensibly saw the non-defense oriented moonshot project responsibility transfer to the NSF, although the funding shift wasn't proportional.
The order of creation isn't exactly relevant to how responsibilities have shifted.

It's kinda like how, for the longest time, presidential security was handled by the Treasury department. It wasn't because presidential security was considered a financial matter, but because that's where it fit.

https://www.darpa.mil/news/features/arpanet

Secure communications and information-sharing between geographically dispersed research facilities were among the ARPANET’s original goals.

From your link to the arpanet wiki:

Building on the ideas of J. C. R. Licklider, Bob Taylor initiated the ARPANET project in 1966 to enable resource sharing between remote computers.

Sutherland and Taylor continued their interest in creating the network, in part, to allow ARPA-sponsored researchers at various corporate and academic locales to utilize computers provided by ARPA, and, in part, to quickly distribute new software and other computer science results.

There's a big difference between ARPA funded labs and general university usage.

I'm not sure why it would matter that you worked for them in the early 90s. That doesn't exactly give you a privileged insight into the creation of ARPANET.