Yes correct, but as far as input-to-screen latency is concerned, it is all local and as such it has little to do with network latency. What is important is the perceived responsiveness of the game. For instance, in Gears Of War (not sure if all versions did it), it is usually very hard to tell whether there is network latency or not. The only times you can really tell is when you do something and the outcome is completely unexpected or if your position jumps around (you enter a building but a second or two later you are outside again).
Not sure who here is old enough to remember but in the original Doom games for instance, the way it worked was that any input had to go through the server round trip, meaning that when you rotated it would usually lag half a sec or so, even on a good day. This caused a lot of people to get motion sickness. This was until they changed it to do movement locally.
you can't have both a local arbitrator and a remote arbitrator. All networked games have to deal w latency. When you don't see the problem, it's either low or consistent latency, or you're just lucky and local predictive actions happened to hide the lag. When you notice it, there's lag thatthe local gamecould not overcome.
Im not sure about the original doom game. I've only played it on a LAN and never had a problem