lets just say that i dont like the snes and gba versions very much
and that they tend to illicit a lot of swearing from me

id rather play the 64 version or double dash or the ds version

though i loath the wii version the most

Shift 2 is more of an arcade-sim hybrid than pure arcade, that's why I didn't suggest those kind of games.
I think games like Shift 2 and Dirt 1 appeal more to sim-racing gamers.

Hot Pursuit was the one where you could play as a cop chasing down illegal racers, as well as racers trying to avoid the cops. So you had two different campaigns basically, one as a cop and one as a racer.
It also had a free-roam mode where you could just drive around.
The game was fast-paced and could get frustrating because there's lots going on, you need to avoid traps or set them yourself (emp charges etc) as well as manage to drive well enough.

NFS Undercover is like an earlier version of NFS Most Wanted from 2012.
Its open-world where you could casually drive around and do nothing, or do missions.
It also attempts a storyline. To be honest the storylines in the NFS games have never been that impressive, they seem very inspired by 80's and 90's action films. Nothing exceptional, but entertaining enough for what it is.

You might be thinking of NFS Prostreet perhaps ? many people didn't like it but I thought it was good.

I enjoyed NFS Most Wanted (the older one) and Carbon a fair bit. They are a bit old now, but that's probably alright if it's arcadey you want. I think I prefer Most Wanted overall, but Carbon had some things going for it too (no drag races, muscle cars that arrive everywhere sideways, drifting competitions).

GRID is a lot of fun, too. If you keep stability control and such on, it behaves like an arcade racer; with everything off, it's a semi-realistic piece.

Sonic Racing Transformed fans here as well :D

The next racing game in my wishlist is Distance . Very arcadey with a visual a la Tron style. It's evading hazard, run across checkpoint with limited time, and doing trick across track racing. The car's flying too! Check it out here https://www.youtube.com/watch?v=xViRtuWSGr4 it's still early access though, but very promising.

Some good Oldschool stuff:

Mario Kart

F-Zero

Midtown Madness
https://www.youtube.com/watch?v=Fbj9rg62ep4

NICE 2
https://www.youtube.com/watch?v=-uEye80v4UU

Ignition (aka Bleifuss Fun or Fun Tracks)
https://www.youtube.com/watch?v=uCi4ZSnbJKg