There are really two types of F2P models (possibly more). One where by paying you can actually unlock things that affects game play. For example, League of Legends. You can pay to unlock champions and that obviously affects your gameplay. Or, Eagles in Angry bird. You can use an Eagle to get over a difficult level.
There there is - TF2 and Dota2, F2P model. Where you pay to buy cosmetic items, but it does not affect game play one bit.
I would say a 3rd type are games like Dead Trigger and most of the "-Ville" type clones.
In those games, you can play without spending real money, but it's going to suck and not be as fun. These are the types of games that have the most backlash because they just seem so "greedy".
In games like League of Legends and Angry Birds, you don't need to spend money to get the real game experience. You can still play just fine as a free player and spending money seems more like a bonus instead of a necessity.
I would add another one with builder types of games. You can go about the game very slowly without IAP, or have fun and build things quite rapidly with IAP. Triple Town falls in this category. IAP becomes a continuous purchase.
In contrast to arcade games like Mega Jump, Jetpack Joyride and others. Where you can do pretty well with a couple IAPs that will last throughout majority of your time playing.