As you're from abroad I can understand why the Slade doesn't register so high in your opinion. It's an English thing, a bit like Oxbridge. Most people know of the Slade and recognise the artists that went there. The school is more conservative than most (if conservatism is possible in the art world) reflecting modern British values.
For starters Goldsmiths/CSM have an entirely different approach to student application to the Slade. They are riskier art schools. They gamble with who they admit and consequently create a lot of failures but also as the YBA's have shown (although Saatchi is entirely repsonsible for this*) a few successes. Thats why they are rightly percieved as the edgier art schools. But being edgy isn't everyones cup 'o' tea. For me edginess is fleeting and reserved for the select young few, the rest can fight over the scraps.
I think its important to understand the art world in relation to the 'real world'. The Slade with its alumni and traditions registers highly in peoples conciosness; when applying for a job in the creative sector (which I presume most people will end up doing seeming as fine art isn't for most people a sustainable buisness) the Slade is an instant 'Aha! a creative individual with some intelligence' (note the ABB requirement/selective policy) while imo Goldsmiths is an 'ah, an out of control performance artist, risky...' (thinks Emin, Hirst THIS documentary
https://www.youtube.com/watch?v=KIa-NtGV-ZM). In short it has a much better real world reputation than other schools.
However prejudices and reputations apart, the Slade is better equipped than most schools for fine art. It's in a high ranking zone 1 university with intimate class sizes. This makes it less insular and more open to other ideas from science etc. It also means you have more studio space, better oppurtunities to make closer friends and a great location to explore the metropolis. It has great studio equipment, good links to galleries and wonderful architecture. If you want these things then apply for the Slade.
*Personally I can't detect any movements coming out of the art world, to quote Grayson Perry 'we're living at the end of art...there is nowhere left to explore'. The YBA movement was simply a marketing campaign led by Saatchi to line his own pockets and make him a household name. EMin, Hisrt et al. are all great buisness people and importnat figures in art history and the art world but fundamentally aren't boundary pushers largely because there aren;t any boundaries left to push!
If you want fame, revolution and edge then go to Goldsmiths/CSM if you want security, a career and an education choose Slade.
Hope this helps