Kat, I don't think the cyber-age has anything to do with it. You achieve academic success by studying a subject until you are of the same standard of knowledge as the leaders in the field, are yourself coming up with new concepts in that field, and are known (by the ppl involved in that area of study) to have knowledge in that area. The Internet doesn't come into it at all. As Wolfgang says, there were many self-taught scientists in the 1800s.
Degree-wise, a BS/BEng/BSc/whatever merely says that you've been taught the skills to do stuff competently in your area of work, nothing else. Academic success doesn't start until PhD level, which is the point where you're actually making a contribution to the knowledge in your field.
And academic success isn't the same as base talent. I know ppl who've played guitar for 30-40 years, but none can play like Mark Knopfler, simply bcos they weren't born with the skill. Rather (I think) academic achievement implies research. So a talented painter hasn't achieved academic success, no matter how good they are. But someone (of whatever level of skill at painting) who's done detailed work on the chemical composition on paint over the years, and who knows how these paints were made and how they change over time, that's academic success.
I think the key is the "based on formal study" part of the definition. You can do formal study on your own; you just have to apply the same standards to your research that would be used by "professionals".
I don't think the Mudcat's an institution of learning, though. It's more an institution of _information_. You don't get a degree from a library, you get a degree from a university or college, where you study methods, use those methods in research, and use the library as a source of information for that research.