Does it become a run out if other fielders do it, or is it still a stumping if the batsman gets down the pitch and for some reason the keeper doesn't collect it but let's say the slip fielder collects and breaks the stump?
No, a batsman can only be out Stumped when the wicket is broken by the wicket-keeper, according to Law 39 (Stumped)
The striker is out Stumped...when his wicket is fairly put down by the wicket-keeper without the intervention of another fielder.
If the slip fielder breaks the wicket with the batsman out of his ground, it is a Run out, as defined in Law 38 (Run Out)
Either batsman is out Run out if he is out of his ground and his wicket is fairly put down by the action of a fielder.
The word stumping is only used when the wicket keeper breaks the wicket right after a bowler delivers the ball. When anyone else other than wicket keeper does the same in a similar situation, it is called run out even though the effect for batsman is same