I think I did define it in my post above - ("the percieved tension in the strings as you pluck and how raising and lowering the strings changes that"). To be more precise, lower the strings for less perceived tension, and raise the strings for more. I'm not going to reread this thread again, but I believe you don't have bridge adjusters? I think if you get them, you might find out what we are talking about, assuming you play jazz style pizz.
No, you don't explain What it is, you only say When you perceive this elusive tension (which is not the tension in the string), and how it varies.
I've gone back to Chris' descriptions and I think I know exactly what he means. If you pluck a string, the amount of resistance (Force) the string gives before your fingers release it, and Distance they travel before the string is released, depends on the clearance between strings & fb. A higher clearance invites better digging, which makes plucking harder and more tiresome, but it gives a better defined, stronger sound (as Chris says "bigger and more "pointed" on the front of the note"), while a lower clearance is lighter to play, but gives a weaker sound (more like a bg). And there's a personal choice to be made here (how do you want to pluck your strings on this bass?), which is very closely related to your set-up, depending on personal preferences (including how tired you are). The preferred height also depends on the string tension and elasticity, but of course those do not change if the height is adjusted.
The product of Force x Distance is the Work your fingers have to deliver, and how quickly they do that determines the Power they're delivering, and how long it takes before your underarm start to hurt and blisters appear. Of course the Tension in the string, and the elasticity of the core, but also the string clearance, determine the Force that your finger feels from the string & have to deliver.
I have had (copper) adjusters in 2 bridges and I have removed them from both (new feet). They did not make me perceive tension, but they did make me perceive what I describe above.
Thinking it over, I can see why you would want to be able to adjust this clearance easily (particularly indeed when getting tired). So far I have taken the approach to set the instrument up the way I like it (I have several bridges for each of my basses), and then just played it the the way it is. I can do this because I stick to the same strings, and because my humidity stays largely constant. But I can see why you could get hooked to easy adjustability - because I too get tired.
If you want easy adjustability of this set-up component, however, adjusting the fb might be preferable to adjusting the bridge height. Adjustments could be done by just a single nut below the end of the fb (with a trussrod, or maybe just a tension/compression beam to the heel), and you can do them without (significantly) re-tuning the instrument, so it can be done a lot quicker and you don't need to worry so much about bridge orientation. You may also want to adjust your fb after significant bridge height adjustment. Potential downsides could be: extra load on the fb-heel glue seam & different fb sound (particularly when slapping, but there this issue does not apply anyway?).
Does anybody have any views if adjustable fingerboards on double basses would be popular? Has it been tried? Should this get a dedicated thread?