There's not really an exact date, but I suppose Aug. 6, 1991 will work as good as any, since it's when Berners-Lee first promoted it.
The "Internet" itself is much older than even its name, and much more than a single protocol running on top of it (such as HTTP). It was conceptualized back in the 1950's or earlier, and has been accessible in some form since the late 1960's to early 1970's. I ran a BBS system back in the 1980's that had an Internet gateway through FidoNet, which was still quite a bit before the web.
Realistically, the real "birth of the web" didn't occur until Netscape commercialized its web browser to the masses back in 1994. That's when I, like many others, first started using it from home (although I already had some Internet access at work and through a BBS I ran on dedicated phone lines).
I first tried to register "stardust.com" back in Jan. or Feb. 1995, but just BARELY missed getting it by about a month. So I ended up registering "sdust.com" instead and later got into a lovely trademark dispute with "Stardust Technologies" that required retaining a $300 an hour intellectual property rights attorney to bitch slap them into realizing I had prior use. In any case, the first ".com" domain names were registered back in the mid-1980's.
The Internet has a client/server based architecture, with different protocols running on different ports. HTTP typically runs on port 80, HTTPS on port 443, FTP on 45, SMTP on 25, POP on 110, etc.
That's the beauty of it. There's nothing preventing anyone from creating a new protocol to run on an unused port, then developing server and client software to utilize it. That's what Tim Berners-Lee did when he created "the web".