Unit Elasticity Calculation Question
You have to use the midpoint method to resolve this (if I recall correctly).

https://quickonomics.com/how-to-calculate-price-elasticities-using-the-midpoint-formula/

The reason I may not recall correctly is because the second you introduce calculus into the study of economics, you discard these formulae entirely and (try to) forget they ever existed.  You seem to already know calculus, which is probably causing you to be even more confused rather than less, and understandably so.

The primary purpose of these non-calculus formulations of elasticity is to teach you the intuition around what elasticity is as a concept - I definitely have strong opinions about how well it accomplishes that. :)

EDIT: Just wanted to add that, if indeed the problem is as you're saying it (come hell or high water, Billy is going to spend \$10 and only \$10 on gas) then your intuition is right - Billy has unit price elasticity of demand for gas, although in a kludgy sort of way.  This is supported by your calculus-based analysis.