in terms of the context of your sentence: AT would be talking about the starting point of something. IN would be talking about from the starting point and then for an initial period or duration. So IN is better to use for your sentence: 'IN the beginning it was possible to process information with computers and the term 'data processing system' was used to describe different types of systems.'

Hmm. That's a difficult question. I would say that AT refers to the starting point and IN is more like during but to be honest I don't think they would convey much difference in meaning. To me they seem pretty interchangeable.

A better word would be "Initially", which means something that happens at the start. Initially it was possible to process information etc. In the beginning and At the beginning are not really right in this context.