The heart of the debate lies in the simple fact that there is strict definition of what RPG actually means. Literally, it could be interpreted as a game where you play a role, a definition that would include any story driven game ever made. This is obviously not a terribly useful definition.
In a less general way, an RPG can be considered as a game that allows for narrative agency and/or character agency. Before anyone hits the "angry flame" button, let me take a moment to describe what I mean by that.
Narrative agency means that I, as a player, have some measure of control over the plot beyond simply determining if it advances or not. Most games you'll find do not offer much narrative agency to the player.
Character agency means that I, as a player, have some control over how my character or characters develop. This can take many forms. It may be that I can control a series of stat systems that determine what my character can or cannot do, or perhaps simply determine what my character is proficient at and what he is not. This can take many forms, from the management of complex interelated systems (Such as the character System in D&D), a simple version of the same where options are divided into a series of distinct choices (like in Diablo 2 or World of Warcraft for example), or the management of items and mcguffins that alter the way the character can approach the world (Dungeon Siege for example). It could also take a narrative turn where I can control how my character responds to the world. Note that this only counts for something if the world is capable of changing in response to a change in it's perception of my character. The most recent notable example of this would be Alpha Protocol, but I suspect few actually found anything laudable in that game.
You'll find that such a definition would generally include the various opinions people have about what makes an "RPG". The trouble is, it is still a wide enough definition that it includes a surprising number of games and worse still, it excludes certain games that some people will insist to the bitter end is an RPG just because past examples were RPGs as well.
If we can assume that the given conditions are at least fairly close to the truth, we can examine several games to see how they stack up.
Modern Warfare 2: The player is given no narrative agency. The plot will progress in the same way regardless of choice. There is no character agency either as any choice that would alter the character in a significant way results in a fail state (that is, the player can simply give up and not play any more). As such, I can assume that MW2 is not an RPG.
Mass Effect: The player is given a degree of narrative agency. At several points throughout the story, the player is allowed to make choices that, in some way, alter the story being told. The beginning and the end are, however, generally fixed as is the path the player will take and as such the narrative agency is limited. The player is allowed to choose, in a general way, how to approach a situation. In some cases, this choice of tone will alter the narrative. The player is also allowed to pick from a limited set of skills that give some measure of control over how the game is played but the differences are often slight. The player is also allowed to alter the gear of their character, but the choices generally amount to replace X with a better version of X and thus the value of the choice is rendered moot from a systems perspective. Given that Mass Effect has both narrative agency and character agency, the game could be correctly called an RPG.
Final Fantasy 13: The player is given no narrative agency. The plot is strictly linear with no significant variation allowed. The player has only minimal character agency. While the precise route one takes with the characters will vary, the end result is the same regardless of choice. The only way to alter either character or narrative requires the introduction of a fail state (the player can choose to not finish the game). Final Fantasy 13, in spite of the very minor character agency, is probably not best described as an RPG.
Mass Effect 2: Generally the same as the previous game. Narrative agency remains the same though character agency shifts to a degree. A player has fewer options for skills but those choices tend to result in a more distinct difference in how the game is played. There are fewer items but the choice is no longer between x and better x, but rather between x and y meaning the choice often impacts how proficiently a player can approach a given challenge. Because a player has both narrative agency and character agency, Mass Effect 2 could correctly be called an RPG.
