I don't think Nikon's are using BL if in P mode (maybe with an external flash set to BL and not using spot metering).
Yes, it does. P mode is not different. For any TTL flash without the overt TTL/TTL BL menu (the SB-600/SB-800/SB-900 have this overt menu), the flash will default to TTL BL mode. Including P mode. The menus are abbreviated TTL, but the action and result is TTL BL (meaning balanced flash, meaning reduced flash due to bright ambient). Nikon is a TTL BL system, and a good one, but they have made some of it be confusing and arcane.
That TTL BL mode default includes the internal flash, the SB-400, SB-700, and the Commander with any flash. It includes many if not most third party flashes (those doing iTTL without overt TTL/TTL BL menu), but I do believe there are few exceptions, depending on firmware level they report to camera). The camera is what does the metering modes, the flash just responds to orders.
The only exception to the TTL BL above is when the camera is set to Spot Metering, which always changes the iTTL metering mode to TTL (never TTL BL in any case). The flash does not do Spot metering, but Spot is only about the spot, and the concept of balancing flash to ambient loses all meaning.
P mode is certainly Not an exception regarding TTL BL mode. P mode and TTL BL flash mode is point&shoot fill flash in bright sun (i.e., a little more aware of the situation than is A mode, where we idiots can set an aperture that can make it impossible

).
I understand your point about using the word "primary." I simply meant it will use as much as needed without regard to anything else.
I can agree with that definition.

Balancing flash with ambient that is 6 stops down has no meaning either.

But when ambient becomes significant level, then ambient remains primary, selecting the camera settings used. The TTL flash matches those settings with power level.
In P mode the camera knows it has full use of the flash and will therefore tend to choose settings based upon the flash as the available ("primary" consideration) light source (i.e. it tends to choose lower ambient exposure/"better" camera settings). If outdoors and the camera can already use the "better" settings then it's of no affect. But TBH, I don't use P mode with flash so I'm not 100% certain...
Disagree, and I'm puzzled what gives that impression? What P mode does is to set both shutter speed and aperture based on ambient. So it does know about and honor Maximum Sync Speed (same as A and S mode), and will not exceed sync speed in bright sun. But it can control aperture too, so it won't try to use f/4 for fill flash in bright sun which cannot work.
There are only three differences in P mode and A mode, and two are about aperture and flash.
1. P mode can change aperture, and of course sets both shutter and aperture. Where A mode only sets shutter speed for the aperture we manually select. Both cases are in regard to the ambient, not the flash (except the Minimum and Maximum shutter noted - but flash exposure is not affected by shutter speed, but ambient of course is).
2. Except, P mode introduces a Maximum Aperture with flash, which numerically varies very slightly among models, but maximum is either f/4 or f/5 for hot shoe flashes, and ballpark of f/2.8 or f/3.5 for internal flashes. Further this maximum aperture stops down even more, 1/2 stop for each stop of increased ISO - so at high ISO, the maximum aperture might be like say f/8 (flash in P mode). The manuals for higher models have these charts, the lower models probably don't. That is really not about the flash, they say it is about ability of the metering system to handle those EV numbers (however I am unable to say why A mode is different, or why internal flash is different). But even if you bought a f/1.8 lens, P mode will never use that aspect with flash (the other modes can).
And 3, in Auto FP, HSS normally kicks in when shutter speed exceeds the trigger speed. That is the Auto aspect of it. In camera A mode, if using HSS in bright ambient where the shutter speed is very near trigger level, slight variations (passing clouds, etc) can therefore go past the trigger, and kick FP in or out, causing unexpected huge changes from one second to the next.
This possibly might be behind your thinking that P mode chooses better settings? Because P mode is different, it can change aperture, and will - the lens will first stop down all the way, to f/32 if it can, before it will allow PF HSS to kick in. You can spin it in, but it won't go willingly.
A mode cannot change the aperture, so it can not do these actions.
But there are no other differences in P mode, and none in regard to TTL vs TTL BL flash.
The only thing forcing TTL mode over TTL BL mode is either Spot metering, or an overt menu on the flash. True in any camera mode, A, S, P, M.
I think we've gone well beyond the basic question initially asked which is why I didn't expand on my comments originally... This can all get very confusing very quickly.
I agree.