You'll get no argument from me that it presents a different kind of experience. But as before, it's up to personal interpretation just how much of a better experience it is. I can't attribute immersion exclusively to firstperson because for that I would have to forget everything else that contributes to any immersion I might feel in videogames. There's been a handful of firstperson games which created no immersion for me, as opposed to thirdperson games which somewhere along the road had me refer to the character I was playing as 'me'.
"How the player reacts" seems to be misleading. I am nearly always reacting to things in a game, sometimes as if I would react in real life. I've encountered no short amount of situations where I forget myself and become so enthralled by the gameworld that I forget myself, and oftentimes there are situations where I reacted the same way as if I would act in real life, regardless of whatever perspective is being used by a game. I've been somewhat frightened, wheter by explicit or implicit danger, in X-Com: Terror From The Deep and Aliens vs. Predator. I've felt marvelled by 'my' surroundings, to the point of gawking at them, in games like Torment and Morrowind.
But perspective never influenced me on this, in regards to feeling like I'm there or not. It influenced how I looked at it, not how I felt about it.
Then again, there are different levels of reactions. A gameworld which relies on physical reaction by the player will no doubt allow for physical expressions that become analogous to those we might have in real life. An isometric version of Doom that relies on character targetting for the purpose of attacking (similar to Fallout) won't allow me to shoot as freely as in a firstperson Doom, sure; but the very act of clicking frantically to kill whatever is headed our way exists in both situations... And could probably be done in the first scenario as well.
It could be argued that you never experience anything because games are always about someone else other than you. This is part of why immersion rings differently for me. Experiencing something from a different perspective rarely has an effect on me because the same situation would still have the same value regardless. Watching my avatar burn trough my visual perspective is no less terrifying than seeing it burn trough someone else's perspective because they are both happening to your extension in the gameworld.