That's strange then, I've only heard of dial up being invented in 1996!
In 1995 I had something similar to the image below (USR Sportster 28k) which used to plug into the side (and stick out) of my laptop. What you see is only part of it, as I also had to have an adapter for the laptop. Most modems prior to 1996 were external modems and I got my first PC internal (33k) modem card in March 1996 as a birthday present from my brother.
My first ISP was U-net, a North West Company whom I believe started providing Internet access in 1994, but some others which were also around previously were Pipex & Demon. U-Net also had direct access to JaNET which has been around since the early '70's.
U-Net used to charge £12 per month for 'up to' 56 kbps connectivity then you paid 10p per min peak and 1p per min off peak for each and every minute you used the Internet. It was very easy to rack up phone bills of £100 per month if you weren't careful. U-net was later taken over by VIA networks who failed to keep abreast of DSL technology. A few of their staff later ended up at Zen.