Having now seen significantly more of the PHB and all of the rules covering stealth, hiding, vision, etc., I have some thoughts.
First of all, it really is the Player's Handbook, huh? The actual rules are pretty sparse, and there's almost zero guidance for a DM running the game! A lot of stuff - not just the stealth rules - is handwaved with "the DM decides when you can do this" or similar. That's no bad thing, but it does read more like the rules you get in the starter sets than the definitive rulebook of the game. The idea is that the DMG will cover the other side of things, I suppose, but that's a slight departure from how things have worked in the past, where the DMG was kind of optional. In other words, we're working with half the information here.
I was surprised, also, to find there was a sidebar on vision and being unseen, with the same rather vague rules about advantage or disadvantage, and targetting a space rather than a creature or object. It's very brief, and my assumption that the Invisibility condition was an attempt to codify the "common sense" approach that the 2014 rules went with (less than successfully) appears to have been incorrect. We're still relying on the natural language of "seen" and "unseen", which means that the "unless they can somehow see you" line means precisely what it seems to: you're just meant to apply common sense to adjudicate whether a creature is invisible Invisible, or just not currently visible Invisible. Which is perfectly workable, as everyone was saying back on page 1, but does seem to slightly defeat the purpose of having the Invisible condition, or at least letting it be gained from the Hide action.
What I believe the purpose of the condition is, is to remove the slightly finnicky interaction between Hiding - or making a Dexterity (Stealth) check - and being made invisible from a spell. It was always a grey area, with most people interpreting the rules to be that being magically invisible essentially made it so you could always Hide, but you still needed to take the action, or at least make a check to move stealthily or something, or you could be detected by the sounds you made, etc. This is somewhat counterintuitive, and with Hiding basically just granting the same condition the spell does (albeit in a way that is easier to lose), it removes the need for an Invisible creature to also take the Hide action. Also, Invisible subsumes Surprise entirely, so the assumption seems to be that, when an ambush is being laid, one side will all take the Hide action and therefore be Invisible, thus gaining advantage on Initiative.
Now, what I'm hoping we see in the DMG is some more concrete guidance on how and when to apply these rules. Stuff like how Hide works in the context of the exploration rules, where it's basically glossed over in the PHB (this is pretty important, because Surprise and ambushing as a concept hinges on this!). It's probably too much to ask, but it would be good if there was some discussion on different ways to adjudicate stealth - like whether someone keeps Invisible if they sneak out and slit a guard's throat, for example, with recommendations of when you might want to do it like that (no one else around, sneaking into a secure area), and when you might go by a stricter interpretation where breaking cover = losing Invisible (the middle of a chaotic battlefield).
Tl;dr: The PHB is vague as heck, actually, leaving most things up to DM interpretation. Fingers crossed for more guidance in the DMG. We have wasted over 100 pages arguing over rules we only have half of.
