When Jobs first announced the iPhone, he really sold the idea that it was running a real web browser like on a desktop. Up to that point, there was a special mobile internet that really sucked.
Maybe depends on the country but 00's in Sweden on the internet was OK, lots of social places and other nerdery going on that was easy to get into, granted you had an internet connection. LunarStorm (originally "Stajlplajs" I think) must have been popular around 2002 already, and I remember that from my teenage age group, most of them were signed up to Lunarstorm, even though most of us were still on modems, 240p webcams and white/yellow desktop computers.
You could run things like IRC clients, dedicated text chat apps, and server rendered browsers on live Internet. But downloading full webpages was too much for the hardware.
https://en.wikipedia.org/wiki/Wireless_Markup_Language
> Building on Openwave's HDML, Nokia's "Tagged Text Markup Language" (TTML) and Ericsson's proprietary markup language for mobile content, the WAP Forum created the WML 1.1 standard in 1998. WML 2.0 was specified in 2001, but has not been widely adopted. It was an attempt at bridging WML and XHTML Basic before the WAP 2.0 spec was finalized. In the end, XHTML Mobile Profile became the markup language used in WAP 2.0.