I\'m not a Genre expert TomN, but I\'m probably one of those closest to being an Adventure game expert on these boards so I\'ll try to answer you questions as well as I can.
Adventures, pure adventures, have gameplay that revolves around a few key points.
Exploration. - Walking around a described, prerendered, or fully 3D world and interacting with it.
Dialouge/Storyline - Old style adventures are starting to be called IF or Interactive Fiction. They have a heavy emphesis on storytelling and dialouge between characters. Most of the time (but not always) you are limited to one playable character who you guide through the Adventure.
Puzzle Solving. - This is the real flesh of an adventure game. This is not like Tomb Raider\'s "Find a key, Find a keyhole, Pull a lever, jump, jump, roll" kinda puzzle solving. This kinda puzzle solving includes Item Hunting where you solve a puzzle, gain an item, use that item to solve another puzzle.. ect ect.. and it usually includes a great deal of non-linear or "Non-Logical" thinking. For instance, to save you cheif engineer from the brig in Space Quest 5, you have to get a packet of SpaceMonkeys. On the back it says, do no mix with alcohol. So you have to put the space monkeys into your drink, which then fill up the spacestation with lil green critters. That draws SpaceCommand officers away from the brig, allowing you access to Cliffy\'s cell. Then you have use you pet face-hugger spike (as seen in the picture above) to take a piddle on the bars. The acid in his urine eats through them and lets Cliffy escape. It\'s actually a lot harder than it sounds when you have to figure it all out for yourself.
No or very little Action - In adventure games, there is very little action or fighting. If there is, there is usually a way around around it or a way to avoid it if you want.
There are two reason why, I feel, that adventure games have never been big on Consoles. First off, they\'re brain twisters with very little to no action. Console gamers have grown up with itchy thumbs and enjoy alot of fast paced action in they\'re games. This won\'t get you through an adventure where things are usually pretty slow paced. Another reason is that in the past, Adventure games have almost always been ported over from the PC and built with a KB & Mouse in mind and not a joystick. It makes the game engine cumbersome to scroll through item menus and options to connect actions with items or worse.. a slow moving mouse cursor controlled with the D-Pad. Now that we\'re seeing more and more adventure/adventure based games being build specifically for consoles, they\'re starting to gain in popularity.
As far as what marks a game into a certain genre. It has to do with just how heavily it relies on a certain gameplay aspect. Resident Evil is adventure in my (and many other people\'s) mind because the focus is on item-hunting/Puzzle solving first. You can run past zombies.. and there\'s very few areas where you absolutely MUST kill monsters to advance.. but just see how far you get when you decide not to pick up and use items to solve puzzles. Some games, like Zelda and QFG, defy classification. QFG being my example because I know the most about it.

In QFG, you cannot advance without solving puzzles OR increasing your stats. Many puzzles rely on decent stats in order to complete. Like for example, to solve the healer\'s ring quest.. you have to have either a high throwing skill or a high climbing skill to get the ring from out of a birds nest. Yet, combat is almost completely optional... even for a fighter class character. You increase skills through practice.. not slaying monsters.
Anyhow, if you want to play through a few good "Pure" Adventure games.. then I suggest picking up:
Grim Fedango
Sam & Max hit the Road
Monkey Island
Leather Godesses of Phobos
Adventures in Serenia
Mystery House
Mist
Frankenstein: Through the eyes of the Monster
Gabriel Knight
Kings Quest
Les Manly: Search for the King
Leisure Suit Larry
Quest for Glory (A good example of a hybrid adventure game)
Indiana Jones: Fate of Atlantis
Manic Mansion/Day of the Tentical
Willy Beamish
Legend of Kyrandia
Heart of China
Rise of the Dragon
Zork
Stupid Invaders
The Longest Journey
Riven
Lighthouse
That\'s just a few off the top of my head.
http://www.MobyGames.com will be a lot more helpful. In most cases, you\'ll have to buy these games.. but there are other methods to getting what you want if the game you want to play is no longer sold or supported by it\'s parent company. PM me if you need further guidance.