The specular input of a principled BSDF is not the same as specular color. Principled's specular is much more akin to a remapped Fresnel or IOR. The specular color of a principled BSDF is derived from the base color instead.
In earlier versions, Blender baked only by pointing a camera down the normal of the sample. This mostly makes Fresnel, a view dependent effect, irrelevant to bakes.
In current versions (4.0.1 or around then), the specular settings have been renamed to IOR level which should dispel some of the confusion, and you can now bake from an active camera rather than above surface in order to get view dependent effects like Fresnel:
As Blunder offers, if your goal is just to bake some color, just bake it as emission-- that's the best way to get exactly what you want, when you know exactly what you want.