Pie: a cut above

Years ago we were in a restaurant--I can't even remember where--and it offered pie for dessert. The price was "$1.50 per cut." (We see the same phrase in Zagat-rated restaurants and in the menu for the Pan-American Exposition of 1901, so it seems to be nothing new or unusual.)

The phrase confused me. If pie is $1.50 per cut, does it mean that if you are the first person to ask, for example, for a piece of a pristine raspberry pie, one that had emerged fresh and fragrant from the oven and then taken the air on the windowsill so that it was still warm enough to gently melt the vanilla ice cream, you would be charged $3.00 for your dessert? After all, it takes two cuts to get that first piece. Or is the word "cut" a synonym for "slice"?

I didn't want to ask the waitress, because there are Scared Straight inmates less formidable than the server we had that day. The restaurant must have been in New Jersey, come to think of it.

I couldn't find anything in the Dictionary of American Regional English to clarify the question, so I hauled out the compact edition of Oxford English Dictionary. That was no small feat, because even in the two-volume form, it takes six men and a boy to maneuver it out of the bookcase. And forget about using just that magnifying glass. At my age, I needed my glasses, the magnifier, and the harsh, gynecological light of the bathroom to read the definitions. ("What are you doing in the bathroom, P-doobie?" "Reading the OED to research pie. What do you think I'm doing?")

Definition 23 is "A piece of anything cut off, esp. meat; a slice." Thus, it would seem that a cut of pie and a slice of pie are the same things. So why not just say a slice of pie and not keep me brooding about it for, lo, these many years?