Yikes do these labels offer opportunities for confusion! I'll address the first 3; the HPS lamp I imagine is characterized by some unit but I don't know which.
Here's what the metrics have in common:
- All are a measure of the 'output' of light. Lumens can be translated to watts: https://clark.com/technology/lightbulbs-watt-to-lumen-conversion-chart/. Mols per second also can be translated into watts: https://www.researchgate.net/post/Can_I_convert_PAR_photo_active_radiation_value_of_micro_mole_M2_S_to_Solar_radiation_in_Watt_m22.
Here's how they differ:
One refers to the area over which the light is spread, a square meter. That's a function of how far the light source is from the surface on which it is incident. So to say that a light bulb generically provides x watts per meter squared is making some assumptions about how the bulb will be used.
Some are specific to a subset of the electromagnetic spectrum and some aren't. Lumens refer to the visible part of the spectrum. PAR covers the portion that plants can absorb and use for photosynthesis. Visible and PAR are not identical, although not coincidentally they overlap heavily. There's a nice picture on this webpage: http://sdhydroponics.com/2012/06/13/par-the-light-spectrum/. Watts are the least specific, covering the entire spectrum, or any electromagnetic energy. Light bulbs are made because we want visible energy, and mostly that's the portion of the spectrum in which new kinds emit most of their energy. Old fashioned incandescent bulbs emit an appalling portion of their energy as heat in the thermal part of the spectrum instead of as light. It's why they are so wasteful.
Just to make it more confusing, the wattage of a light bulb classically refers not to how much light it emits but to the amount of energy (electricity, generally) that it uses. An incandescent bulb that consumes 40 watts emits a predictable amount of light, which is how (and why) the translation to lumens is increasingly made. Watts nevertheless can be a measure of energy emitted, as they are in the PAR label.
In a perfect world one could translate any of these labels into any of the other units, though one would need some information about the emission spectrum distribution for each. Intensity (whether watts or mols) of PAR is what matters for the plants. So that's where I'd focus.
I'd keep in mind that the visible-light wattage equivalent of an x lumens lamp is not far removed from the watts of PAR that it emits.