If I measure the signal strength 1 km away in line with the boom in the direction in which the Yagi is pointing, and in exactly the same plane as the elements, and then compare that to the signal from an isotropic radiator, the yagi will have a signal which is 7.5 dB higher.
Theoretically correct. In practice you'll have some trouble actually performing this measurement. Firstly, isotropic radiators don't exist. Secondly, if the specified gain of 7.5 dBi is in free space, the antenna installed over ground will have different characteristics. The location of the mast or tower on which the antenna is mounted, feedline, and other nearby antennas can also perturb the pattern, sometimes very significantly.
Your other two examples seem to involve in some way integrating the power radiated by the antenna over all possible directions. You'll find most antennas likely to be used for amateur radio transmitting will give about the same total power as an isotropic radiator.
The reason should be fairly intuitive considering the law of conservation of energy: if a directional antenna radiates more strongly in a particular direction, it must radiate less strongly in some other direction. To do otherwise would require creating additional energy from somewhere.
The relevant metric comparing total power radiated in all directions is called antenna efficiency. It's not difficult to achieve efficiencies above 99%. Consider a simple wire dipole: the wire has very low resistance, so there isn't anywhere for a significant amount of energy to be lost: it has nowhere to go but into electromagnetic radiation. A mobile HF antenna has lower efficiency: the electrical shortening increases the current in the antenna, making the resistive losses in the antenna and the loading coil much more significant. A vertical installed without radials has lower efficiency due to the high resistance of the soil. But antennas that aren't shortened, and are properly installed, can have efficiencies that are very nearly 100%.