I believe, at the time of the BB song, that 'short' was a more general term for the sort of modified car, whether a rod or a custom, that a kid might drive. Considering the way slang changes in meaning in various parts of the country, I'd readily believe that Chicago has (had) it's own variation, as Felony has alluded to.
How many of you know that, in the Metro Detroit area, at least, it was the custom to use 'hot' where it's now, and has been for many years, all over the country, the universally used 'cool'. Cool, around here, was a response to hot meaning 'even more so'. "It's so hot it's cool." That would have been in the early-mid 50s around here.
KS