miliWatt to 10.65 dBi Gain

[Alabama Buckeye]

A bit of technical help in understanding 10.65 dBi Gain. As I understand it a 10 dBi gain coincidentally is a 10 times greater power in the direction of the director; provided https://antennatestlab.com/antenna-e...gain-dbi-scale, is correct. That is not 10 times the initial 10W power emanated but 10.65 dBi more than that portion of the 10W which would be received under isotropic radiated power. And perhaps as measured in mili-watts in the sphere? Am I correct in this thinking. Thanks.