I have no idea why you are seeing different color rendition in those two different modes. I don't shoot with Nikon digital (I have a number of vintage Nikon film cameras and lenses). I never shoot with Auto (some of my cameras don't even have it). And I very rarely use Program mode.
You are right about setting a custom white balance. Done right, that will get you the most accurate rendition of the subject, if that's what you actually want. Even when shooting film, we often used "warming" filters or other filters to make subtle changes (usually) to life. Whenever we did that, we weren't really trying to perfectly match reality, anyway.
Today with digital there are Warm Cards that do much the same "tweaks".
https://www.vortexmediastore.com/pages/warmcards-white-balance-system. Those are used to set a custom white balance with a slight bias, which often looks "better than reality". The set of cards have three levels of warming, at least a couple cards for cooling, one for fluorescent lighting, a gray card to set exposure as well as a pure white (to set unbiased custom WB).
You are also correct, the rear LCD on cameras cannot be trusted to evaluate color rendition. It's not really calibrated and is subject to a wide range of ambient lighting conditions that can strongly influence what you see.
Something else you might want to try is shooting RAW. That gives you the most opportunity to adjust WB after the fact, in post-processing. With RAW you are also working with far more color data than what remains in a JPEG.
When you shoot RAW the camera does in fact record the color settings and applies those to the preview image seen when "looking at" the RAW image on the camera monitor or a computer screen. You actually cannot directly view a RAW file... you are actually seeing a preview of it with the color settings applied.
However, a RAW file allows you to freely change the color settings... temperature and tint. (RAW files don't have in-camera noise reduction, sharpening, etc. applied. Those need to be done in post-processing.)
I definitely do not agree with some other responses that say "all professionals only use Manual". That's BS. In fact, many pros and advanced amateurs use M when they can, when it's the correct mode for the subject matter and situation. But there are many times when another exposure mode is required, does a better job. It might be shutter priority mode to freeze subject movement or deliberately blur parts of the image (see panning technique discussion). Or it might be aperture priority when you want to control depth of field effects... blur a background down or make sure everything near to far is sharp. Or it might be M plus Auto ISO (which is actually another auto exposure mode), which allows both to be controlled. There may even be times when P mode is ideal (I usually only use it when I've been shooting in another mode, but need to take a quick shot or two in radically different lighting, then return to the first mode to continue what I was doing.)
Most pros learn to use all the modes of their cameras and when each is most needed or useful. Except full "Auto" mode. That's too limiting. It's basically a "point n shoot" mode, put in the camera for people transitioning from cameras that offered no means of adjustment. The "scene modes" just as sports, landscape, portrait, etc. are also relatively worthless. Full auto and scene modes do a lot more than control exposure method. They also override things like frame rates, autofocus setup, even whether or not the flash will work and the type of image file you can save to the memory card.
I gotta ask... Are you looking at your image on a calibrated computer monitor? And what software are you using to view them? Does it accurately render color?
I looked at your images in Photoshop on my calibrated computer monitor and found that with some slight tweaks I thought the 2nd image... the one shot in P with Auto WB 1.
I felt both of your shots done in P were a little lower contrast. The Auto shot appeared a bit higher in contrast and saturation.
Below is that 2nd image (P and Auto WB 1) with nothing more than a little added contrast and a curves adjustment in Photoshop. I can't view the original scene, so can't be sure this is an accurate rendition. But to me it looks the "most natural".