Well, this all seems "unnecessarily awkward" - especially that part about the too large image thing . . . As god is my witness, when I originally posted that hosted image, that picture was the entire width of the text box. I must have spilled some water on my laptop, and the image shrank? Well, I can't explain it.
Since I know the model number is on the lawn mower (and it's self-propelled, but not a riding mower), I'll try to cross-reference that and check the online manuals. I read some "hose coupling" speculation posted here, but that seems incredibly unlikely. Since I saw this basic same fitting on virtually every lawn mower Sears had in stock, I just assumed that someone would recognize the fitting and have already noodled the functionality. More to follow.