The natural language phrase such as square root of x for a function $Sqrt(x)$ precedes the input with the word of. When we talk of a bivariate function, such as $x/y$ aka $Ratio(x,y)$, we say simply x divided by y or, possibly ratio of x to y.
So, the phrase, price elasticity of demand would imply, not a bivariate function, but a function $PriceElasticity(demand)$.
Moreover, when we speak of causation in mathematical models, we usually, likewise, speak in terms of the future being a function of the past. Therefore, the thing named by the function is the effect; the response of the model. The input to the function is the stimulus.
So this leads one to interpret "price elasticity of demand" as a change in demand resulting in a change in something called "price elasticity".
Lest people think I'm being just a mite too persnickety about all this natural language stuff, there does seem to be a major gap between the economics profession and even the college educated public affected by economic policy, resulting in what might charitably be called "an impedance mismatch".
After looking into this a bit, and discovering that the economists' "elasticity" has little to do with the long history of the word in mathematical physics, I'm wondering if something more along the lines of the bivariate function ratio might better serve both the public and the economics profession.
Or am I simply displaying ignorance of the concepts behind "elasticity" as used in economics?