It seems the reason so many folks think the truss rod adjusts string height is because IF you leave everything the same, in most cases the only reason the action changes is due to the neck moving. So, if you adjust the truss rod it SHOULD bring things back to where they were. Therefore, IF the action is wrong for you ONLY because the neck's relief changed, then the truss rod is the correct way to adjust it.
But, it's a huge disservice to people just learning this stuff to say "almost all of the adjustments to string height can be made with the truss rod." That clearly assumes that the saddles and nut already are in the correct position. Now I haven't touched the saddles on any of my basses in two years (because I've not changed string gauges or brand in that time), and my Precision hasn't had the bridge saddles touched in about six years. But the necks move with the weather and so when the action gets wrong, I check the neck relief and adjust the rod. That solves the problems for me.
BUT it's vital for people to understand how a bass works. The truss rod ONLY adjusts the neck relief. It has a side effect of altering the string height at certain places on th neck, but it doesn't change the saddle height of course. And the problem is that if you don't know what proper relief is for you and your bass, then you'll likely wind up messing things up trying to lower action when the relief is right.
So, when setting up an instrument that's new to you, or with a new set of strings (i.e. different gauge, different material, different brand - don't assume nickle Ernie Ball 45/65/85/105 will need the same setup for you as DR Sunbeams in the same gauges!) check the relief and the nut and get them right for what they do. Once the relief is where you want it for how YOU play THIS bass, then use the saddles to adjust the action.
String measurements are from the bottom of the string to the top of the fretwire. And factory specs are only a guideline for a setup that's going to allow for an instrument to be shipped all over without too much change needed, and to satisfy a wide range of players. Personal setup can be WAY different. For example, I find Fender's specs to be very awkward to play. My basses are set up right now with only about 1/64" or less relief (measured at the 7th fret and fretting the E string at the first and last frets to serve as a straightedge).
John
John