I must point out a couple of things.
1. First about the constant K. Although each meter manufacturer can choose to use a different constant K, I have found that basically only K12, K12.5 and K14 were being used. The difference between K12 and K14 is about 1/6 stop so meter reading differences are not related to the constant K in use.
2. When you have different meter readings between cameras and meters it has to do more with the metering pattern of the cameras and meters. Most cameras are center weighted and meters are not. To compare a surface with the entire surface has the same reflectance (any tone, a gray card is fine) and the surface has to be evenly lighted. The surface must then fill the frame of the camera on entire measuring field of the meter otherwise they won't be the same.
My 2 meters give readings that are withing 1/10 stops of each other. My cameras agree with my meter withing 1/3 stop.