if the game allows it, its not cheating

The only two games that I don't feel bad about "cheating" in are Demon's Souls and Disgaea.  For DS I figure turnabout is fair play and if you didn't want me to hide behind that rock, you shouldn't have left it there.  For Disgaea, I figure it's not cheating, it's home field advantage to pull people into the map, combo attack, return them to base, and reposition.

When Joey Greco turns up with your wife, you're cheating.

When i lose real bad

Using unfair means to succeed would be my definition. 

When I was 7 I used a turbo controller to get through the the torture section in MGS? Did I cheat? D:

I wouldn't consider it cheating what you did in Puzzle Quest. The game gave you the option to play it on that level, but I do think there is something to be said about beating a game on the Normal difficulty. It's like a right of passage. Saying "The game was designed with this difficulty in mind, and I won." So no, I wouldn't call it cheating, but you are a pansy (lol).
 
I also wouldn't nessecarly call what you did in SMT cheating either. it was put there by the developer for you to use. I think you abused it more than you cheated the game. I assume it made the game easier, but within the games rules. So no, I wouldn't call it cheating, but finding an easier way to beat the game.

I beat Punch Out by using the headgear (AKA easy mode) on the final boss.  :p
Anyway, Puzzle Quest is just Rock-Paper-Scissors, so who cares if you cheat?

When I get killed by a dude in MW2, he's cheating.

I just finished up the Xbox 360 version of Puzzle Quest: Challenge of the Warlords (yes, that's the first game, and yes, I know I'm behind on the times).  A clever little game, got a bit dull at times, but enjoyable nonetheless.  It's fun to think that puzzle games like Bejewled are all about luck, and building an entire game about manipulating that luck, and customizing your character to decide what kind of luck you want to specialize in. 
 
Until I got to the final boss, who was easily the most frustrating part of the entire game.  He had a series of spells that make it easy for him to get lucky shots on me, as well as make it even more difficult to use my own spells against him.  I spent a good part of the game grinding up to level 50 to fight him, and he was still no match for me.  Over and over I attacked him, and half the time he would kill me before I even got him down to half health.  I didn't see any way I could win other than pure luck. 
 
Then, one time I loaded up the game, took a look at my character sheet before starting the game, and noticed that I can change the difficulty on the fly.  Easy mode gives you less experience, while hard mode gives you more.  I figured that since I had already maxed my character out, I should try easy mode.  I fight Lord Bane again...and beat him on my very first try.  While he still got incredibly lucky at parts, it honestly felt like he wasn't even trying; several times he had free excuses to screw me over, but didn't.  So...did I beat the game legitimately?  Or is that cheating?  I played through the entire game on normal, and switched to easy for the final boss because that was honestly the only way I felt I could beat the game. 
 
I'm a bit ashamed to say this isn't the first time I've done this; I ran into the exact same predicament in The World Ends With You.  Again, I played through the entire game on normal, only to find myself stuck at the final boss.  The game generously asked me to switch to easy after several deaths, and considering it was the final boss, I accepted.  However, TWEWY and Puzzle Quest are a little unique, in that the difficulty settings can be switched at any time, and offer tangible benefits and penalties for changing them.  More action-based games, like God of War, offer a switch to an easier difficulty as a sort of "mercy mode".  Die enough times in a fight, and you'll be given the option to switch to an easier setting.  However, in times like this, I tend to just stick to my guns and tough it out; God of War offers no way to increase the difficulty back up for the rest of the game, so I would much rather finish the game on my own terms. 
 
Difficulty settings aren't the only way to "cheat", though.  As I mentioned in my last entry for Shin Megami Tensei: Strange Journey, I needed to input a password for a custom-made demon that another player had created.  This was something the developers intentionally put in the game, so that players could share demons with each other, as well as special demons offered as an alternative kind of downloadable content.  But I didn't raise that demon myself; I didn't carefully fuse its skills.  All I did was input a code and grind for the money to buy it so I could finish the game.  As someone who once played Pokemon competitively, that doesn't sound too different from someone buying a gameshark and just crafting their own custom Pokemon.  Then again, this disconnect between players was solved by someone just going out and programming a Pokemon battling simulator for the computer; all the fun of competitve Pokemon fighting, none of the tedious egg hatching. 
 
Still, none of this actually answers the question of what cheating is.  Hacking into a competitve server to kill all the other players who are much better than you is definitely cheating, but for single player games, it's much different.  Then again, people have different concepts of fun.  To some, the desire to see a game to its end is all they care about, while another may prefer the journey itself, and think the trials they go through make the end all that sweeter. 
 
By the way Atlus, making Demon's Souls marginally easier for two weeks may get people to buy the game, but it's still frustrating as all hell.