Not sure what links you were looking at, but no, SIDS is not suffocation. Currently it does seem there is a push for coroners to put the two groups together under one name, and I can't remember what it is, but for when it is suffocation they know it. When it is SIDS it means there is no known cause.
granted, most of what i've read states pretty straightforwardly and without hesitation that bedsharing is safe, and i generally don't expect to be one of those parents who is frequently or highly concerned about SIDS, as i expect our family's shared sleep to be an experience of bonding, affection, and cooing at our newest family member (as opposed to constant fretting). but, for instance, at this link, the way i read it would cause me to conclude that preventing SIDS is all about baby continuing to breathe:
"The proposition that bedsharing is causally related to SIDS is coherent with theories that respiratory obstruction, re-breathing expired gases, and thermal stress (or overheating),which may also give rise to the release of lethal toxins, are all mechanisms leading to SIDS"
aside from the passing mention of overheating, discussions around SIDS (and avoiding risk factors) seem to all point to baby's breathing. even though it may be listed as a distinct cause of death apart from "suffocation", i still get the take-home that SIDS is basically just the baby suffocating, whether that is from the face being physically covered, or suffocating from "gases" or "lethal toxins" or perhaps even just hot, stale air. i get the impression that when we call it SIDS it means the baby most likely suffocated (experienced oxygen deprivation b/c breathing ceased or was prevented) but we don't know for sure, so we call it SIDS. i mean, even if baby overheating somehow stops them breathing, they are still dying of suffocation. but then again, i am probably trying to simplify it and tie a neat little bow around it, b/c that's what people do, even when it's not that simple. and i probably want to tell myself it's as simple as keeping baby breathing b/c then i feel more in control and safe. but it still seems this way (to be about breathing), on the whole, to me.