ARPA was military.

"From 1958 to 1965, ARPA's emphasis centered on major national issues, including space, ballistic missile defense, and nuclear test detection.[21] During 1960, all of its civilian space programs were transferred to the National Aeronautics and Space Administration (NASA) and the military space programs to the individual services.[22]

This allowed ARPA to concentrate its efforts on Project Defender (ballistic missile defense), Project Vela (nuclear test detection), and Project Agile (counterinsurgency R&D programs), and to begin work on computer processing, behavioral sciences, and materials sciences. The DEFENDER and AGILE programs formed the foundation of DARPA sensor, surveillance, and directed energy R&D, particularly in the study of radar, infrared sensing, and x-ray/gamma ray detection."

ARPA was renamed to DARPA in 1972.

"DARPA supported the evolution of the ARPANET (the first wide-area packet switching network), "

https://en.wikipedia.org/wiki/DARPA

There was no Internet in 1972. It was ARPANET which was run by DARPA. The D was added to reaffirm that ARPA was the department of defense, not civilian research.