The developers of ARPANET, the precursor to the internet, varied in philosophies, with "free give and take" being a common one, which does not involve money. And on the other end of the spectrum is BACS, the Bankers' Automated Clearing System, which you no doubt have used many times and which began in the 1960s during this age of ARPANET.
The sub-cultures that you may be looking for may not have been recorded well, but here are some references that gives thoughts contemporaneous to your desired time-period.
Diebold, J. (1967, December). When money grows in computers. Columbia Journal of World Business, pp. 39-46
Reistad DL (1967) The coming cashless society: Implications and benefits of pending system. Business Horizons 10(3): 2332
Bacard, A. (1994, January). A cashfree society: Nirvana or nightmare? The Humanist, pp. 41-42
Bátiz-Lazo, Bernardo, Thomas Haigh, and David L. Stearns. "Origins of the Modern Concept of a Cashless Society, 1950s1970s." The Book of Payments. Palgrave Macmillan, London, 2016. 95-106.
Lauer, J. (2020). Plastic surveillance: Payment cards and the history of transactional data, 1888 to present. Big Data & Society, 7(1), 2053951720907632.