Terminators do feel pain, this is explicitly stated in the above quote from "Judgement Day". Kyle Reese seems to contradict this in the original movie, but he likely meant to say something different (as in "terminators don't care a lot about pain, but I would appreciate if you were a little more careful with my wound") or he might simply not have been aware of it. The terminators that he had encountered in his life (as depicted in the original movie) were more rampant juggernauts than human infiltration units.
However, the terminator's perception of pain is evidently not the same as it is for humans since the effect of wounds are not the same. The "classic" terminator is basically a steel skeleton with servo motors and artificially grown flesh mounted on top of that for camouflage. From a purely functional point of view, the entire flesh coating could burn off (as seen in the original "Terminator") and it will still be entirely functional.
Damage to tissue in a human body severely impacts the human's ability to perform tasks. The reason is that pain is an important survival instrument. Movement tends to cause additional blood loss and additional tissue damage, which is unfavourable for survival. Pain is therefore a very discomforting and disabling sensation.
Terminators are infiltration units. Damage to tissue in a terminator impacts its ability to pretend being a human (with, say, a lost eye and metal parts visible in the hole, or with a cut-off hand and the metal skeleton visible), but does not otherwise affect its functionality. Pain is therefore an important tactical detail to be aware of (which is the likely reason why terminators do feel pain), but not necessarily disabling.
A terminator might change its behaviour based on the received pain data so it does not give away its true nature.
For example, being close to an exploding grenade, a terminator that has not been identified as such might apply a feint and simulate short unconsciousness, and then, being wounded and feeling pain, have trouble getting on its feet again, and limp. When wounds are treated, it might show pain reactions to the human touching the wound. That behaviour may allow the terminator to proceed with its infiltration job and get to its target more easily.
A terminator might "feel" pain in a human like manner and display all its adverse effects until it is close enough to its primary target. At this point, camouflage becomes irrelevant, and while still receiving the "pain" sensory data, it would turn off the simulation, as it is now no longer mission critical.
Indeed, the terminator hybrid Marcus in "Salvation" not only does feel pain, but he is not even aware what he is, or that that he is merely running a simulation (the heart and the brain being the only operational and necessary human components -- yes this is biologically impossible, but alas). It's the perfect deception, the deceiver himself doesn't know about it.
The more advanced shapechanging terminators would presumably have no notion of pain since there is no living tissue from which they could get this data. However, they, too, should arguably be able to simulate pain based on other sensory data such as "I had a hole blown in my leg".
It's what the T-1000 imposing the wounded Sarah Connor does, too.
T-1000 in "Judgement Day" arguably shows real pain when taking its showdown bath in molten metal, although this does not make a lot of sense (other than "dramatic effect"). It is screaming in agony and changing from one form to the other. One might however simply call this behaviour "malfunction".
I'm inclined to believe it is a lifelike simulation of pain due to real damage caused to its structure. From the terminator's point of view, that would be pain as real as it could possibly get.