Hmm... that kind of worries me and makes me think I might be doing the exercise wrong. I have never floor pressed with a barbell, always with dumbells. I have typically done it two ways, either with for 3-4 sets of 6-8 as heavier stimulus after RE bodyweight dips or light bench press or for 2-3 sets of 12-15 as a third exercise on an ME day. When I do heavier reps, I usually pause for 2 seconds with my elbows touching the floor but when I do higher reps I kind of just do touch and go.
I may be using "lockout" improperly. I tend to think of bench in two stages: getting it "off the chest" and "locking it out" once the bar starts to slow down. I know I am over-simplifying things a lot there, but I am definitly not advanced in the bench press at all so I know my understand of the technicalities of the lift are not great.
I will also say that I have a terrible arch (I know floor press takes a lot out of it, but you can still see a lot of guys have a decent arch when they floor press). Add the fact that I am skinny as shit and have no appreciable chest/stomach mass. I also have very long arms, even for a 6'4" guy, so suffice to say that my triceps take a beating no matter what form of pressing movement I do.
So I think the fact that I consider it a tricep/lockout exercise has to do with the combination of my inexperience with the nuances of the bench, my body type and the fact that I always was under the impression that it was for the triceps, so I probably focused more on getting them involved while doing it. I feel like I am performing the exercise properly, but next time I do it, I'll have somebody check out my form to see how it looks.
I definitly appreciate the criticism because it really made me think about what I was doing. If you have anything to add, please be more than welcome to do so.