You are basically correct, with some additional complexity I'll explain.

Terms the game uses, then doesn't define:

Crouch: when your toon is crouched down, either standing still or walking slower in crouch mode. While in crouch your character is always trying to enter sneak.

Sneak: crouching and successfully hiding from npcs and enemy players while moving around. In sneak the eye cross hair will be closed (or mostly closed) and your character will be shadowed and visibly darker. A sound cue lets you know when you're going in and out of sneak. In sneak your player has a detect radius that is significantly smaller (you cannot see it, you just have to feel it out and learn it). Outside this detect radius you are invisible to NPCs and human pvp players.

Stealth: Almost the same as sneak, but basically meaning that nothing sees you regardless the reason. This might be because you are sneaking, or maybe because you've cast cloak, or because you have an invisibility potion.

Invisible: The state of being unseeable for any reason, either because you're successfully sneaking, or because you have some ability or potion.


Things that break sneak/stealth:

-Seen by non-violent creature: let's say I'm in pvp and stalking some enemy reds. They're 50 meters away, and I crouch-walk into view of a mud crab. I hear the exit sneak audio cue, but I don't mess with the controller and exit crouch. I am no longer sneaking to the crab that sees me, but I am still sneaking to the enemies - they can't see me. I am still sneaking to them, but I am not in stealth because something sees me. Attacks against them will be considered crouch: yes, sneak: yes, stealth: no.

-Aggro a violent creature: if I'm in sneak and I wander within the detect radius of an enemy creature, they'll aggro. When a creature aggros, their whole group can see me and I cannot enter sneak until something breaks the aggro. Pvp works differently...if a human enemy sees me, only they see me, not their whole group. They then have to deal damage to me, cast some ability, or use a potion to make me visible to their group if they want everyone to see me.

-Take or deal ANY damage: even one point, and you're out of sneak. In pvp, say I throw down caltrops and go wander off in sneak. If someone hits that and takes damage, I am immediately thrown out of sneak. If I wander into caltrops or any aoe and take damage, I'm out of sneak. In pvp this will make me visible to anyone who is looking - the detect radius in pvp is whoever can see you. In pve, getting thrown out of sneak increases my detect radius to normal. If I'm standing close to monsters, they will aggro. If I'm far away, they might still not see me.

-Use of some abilities: many abilities don't deal damage, like heals and wards. Some break sneak, and others don't. Breaking sneak with a non-damage ability doesn't mean there's an instant aggro - it means your detect radius goes back to regular for a second. In pvp, if I cast vigor, an enemy players will be able to see me for a second, then I'll re-enter sneak and vanish. In pve, monsters will aggro if they're inside my larger detect radius. If they're still outside it, they won't. Test your toons' abilities and know which break sneak.

Damage and sneak/stealth/crouch

The computer assigns an RNG damage bonus to successful sneak/stealth attacks. You get one hit with this buff, then you need to re-enter sneak to get it again. The NB's cloak can expedite returning to sneak, but no you cannot hit, cloak, hit, cloak, hit, cloak, etc, and get a bonus over and over.

The tooltips for abilities and sets often refer to damage buffs under these circumstance. Here's how you get the buffs.

Attack from sneak buff: your toon is crouched, the crosshair eye is usually closed or half open and you're in shadows when you attack. If a non-violent creature sees you, the sneak isn't broken. As soon as your first attack deals damage, the sneak is broken and the buff ends. You cannot stack multiple sneak attacks...just the first counts. To reset the sneak attack buff you have to re-enter stealth, meaning you need to end any npc aggro and hide undamages for 2-3 seconds in pvp.

Attack from stealth: You need to be invisible to all neutral or non-friendly creatures. This is where the "sneak up behind" rule comes into play. If I'm going to melee attack, that means getting really close to an enemy. If they see me before the attack lands, the stealth buff is negated. So by sneaking up behind, I reduce the chance they will break the stealth. However, range attacks like snipe can definitely be launched from any direction and the stealth retained. If I remain invisible, I can launch snipe from 20 meters right in front of an enemy and get the stealth buff. Also, the nightblade ability cloak makes me invisible - this can be combined with melee attacks to get a stealth buff even when slapping someone right in the face.

Attack from Crouch buffs (pretty rare, but some sets refer to crouch): the game is pretty obtuse about this buff. What players infer (which has never been verified) is that the developers intend such buffs to proc only when you are invisible because you are crouched, and NOT when you are standing and cloaked, or standing and invisible for any other reason. Many sets that hit the PTS refer to crouch, then cloak procs them and then the tooltip is changed rather than the set nerfed.