LIAB, the Internet had been around for a while before Berners-Lee proposed the WWW context in 1989. By 1993 that concept was in place and the first public browser was made available (Mosaic). In Spring of that year the decision was made and announced that WWW was free (you still had to have an ISP to connect, of course) and the thing exploded.
Yes, that's what I said. The "internet" as a whole (which compromises of many services like usenet, "the web", email, IC, and more) were around well before 1989.
"The origins of the Internet date back to research commissioned by the United States government in the 1960s to build robust, fault-tolerant communication via computer networks. This work, combined with work in the United Kingdom and France, led to the primary precursor network, the ARPANET, in the United States. A 1980 paper refers to "the ARPA internet". The interconnection of regional academic networks in the 1980s marks the beginning of the transition to the modern Internet." -
https://en.wikipedia.org/wiki/Internet
As stated previously, it was available only to "academics" (educational institutions, scientists, etc) to more conveniently communicate share research. Now the web was introduced in 1989 as you describe. We all habitually refer to the web as the internet, when it's really just one of the services available on the internet.
I don't recall AOL before the WWW as a content provider. Perhaps it was, but I seem to recall that Prodigy, CompuServe and GEnie were the big three in that area. AOL may have been a bit player in that space. But AOL came into it's own when the WWW started in 1993.
Those were the big 3, but AOL most certainly was around. AOL was the first online service I ever joined, but that wasn't until 1995. Their portal to the web was pretty crude though... a very rough web browser that was slow and barely worked. The real appeal of their service then was their own content and services.
EDIT: I should expand on that thought a bit. AOL's growth certainly exploded in the mid-90's, in part because they offered access to "the internet" as part of their service, and while hordes of people joined up because of the growing buzz over "the internet", they mostly didn't know what that really was. To them, AOL and its own services WAS the internet. I should also point out that if people simply wanted to get to the internet, it would have been FAR cheaper to simply sign up with an ISP, which typically cost $12-$15/month flat rate, IIRC. AOL, on the other hand, billed by the minute and could easily cost you a hundred bucks or more monthly. It wasn't until 1997 or so that they went to a flat rate model.
Here's a better read than the Wikipedia entry (EDIT: this is from TIME Magazine's site):
AOL at 30: The History of America Online, Founded in 1985