It is power which leaves the transmitter.

Power is what is received by the receiver.

So the S-meter is a power meter.

It is a great pity that the usual S-meter is scaled partly in S-units
and partly in decibels relative to S9.

But given that 1 S-unit = 6 dB, the modern meter scale fits very
nicely between receiver internal noise level and the receiver overload
point.

And, for example, it's so much easier to report signal strength as S5
rather than 0.047 micro-microwatts.

In the same vein, we could, of course, report a signal strength of 20
dB over S9 as S12 and be done with decibels.
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Reg, G4FGQ.