A watt of heat is a watt of heat no matter the source..
200 watts from a pump, motor, led, halide no matter what it is the same heat. Has nothing to do with efficiency..
A watt in is a watt out.
Mainly light out put has to do with efficiency, there can be some loss in the electronics.
This is the only part of your post that's right.
The reason Lumens was mentioned rather than PAR is because PAR is local brightness (at specific spectra), where lumens is total light output. A lens can affect PAR or another brightness measurement, but it doesnt affect lumens.
All light sources turn some of their electricity into light, but most into heat.
(The following numbers are all best-case scenarios)
Incandescents are ~98% heat and 2% light
MH is ~76% heat 24% light
LEDs are ~50/50
So our theoretical 250w MH is putting out 190w of heat while the 212w LED is putting out 106w
This is why an equivalent LED runs cooler. It's more efficient at producing the light from the electricity it uses.