The definition that includes all three is the most common, widely accepted, and long standing and as I noted it long predates both of us.
This doesn't really concern me much, honestly. For the majority of film's history we had very little understanding about how it worked, atomic theory wasn't even discovered until 1914. So a historic definition really doesn't hold much weight IMO.
The scene illuminance should not be considered a part of exposure because it only determines the *maximum* amount of energy available over any arbitrary time, not the actual accumulation of light at the medium over the duration of the time specified. Because time and attenuation are arbitrary variables not dependent on illumination, and if sensitivity is a factor of serviceable exposure, then likewise illuminations must be considered separately.
Without illuminance no exposure is possible. Consider this definition from the Ilford Manual of Photography which bares an original publication date of 1890, although this is copied from the 6th (1970) edition:
"Exposure
When a photograph is taken, light from the various areas of the subject falls on corresponding areas of the film for a set time. The effect produced on the emulsion is, within limits, proportional to the product of the illuminance E and the exposure time t. We express this by the equation
H = Et
Before international standardization of symbols, the equation was E = It (E was exposure, I was illuminance) and this usage is sometimes still found. The SI unit for illuminance is the lux (lx). Hence the exposure is measured in lux seconds (lxs)."
They don't mention attenuation through the lens in their definition: Exposure is accumulated illuminance over time. They're not concerned in the definition with the camera control factors. So we have this distinction between the hard definition (lxs) and then the functions when taking a photo that produce exposure.
Back to the Wiki definition: "In photography,
exposure is the amount of light per unit area (the image plane illuminance times the exposure time) reaching a photographic film or electronic image sensor, as determined by shutter speed, lens aperture and scene luminance."
Parse the sentence carefully and you get exposure defined as amount of light per unit area (illuminance times the exposure time). That's the same as the Ilford Photo Manual definition.

In the second half of the sentence the Wiki definition provides a list of the three practical functions that can control or alter exposure -- a subtle distinction.
I think we're just dancing around a lack of clarity over that distinction. I'm guilty for mixing the term "definition" up with the practical control functions. My original question did ask for those control functions as opposed to the hard definition:
"Fill in the blanks:
Photographic exposure is a function of _________________, ____________________, ________________"
The hard definition requires illuminance.
In any case it's all good as long as we keep this guy from mucking it up:
Medium sensitivity does not belong to either the hard definition or the list of practical control functions that determine exposure.
Joe