What’s the origin of the phrase ‘Spread eagle’, meaning, to my mind, a person laying on his or her back, with the arms and legs spread out to the side.
…and feel free, if you must, to go down the dark road for an explanation. 
The term first shows up, according to the OED in 1570, used to describe the eagle on coins in Europe.
First cite in 1824 for the figure-skating term.
1785 in the manner which you are suggesting, arms and legs outwardly displayed. It was used about a guy who was gonna get flogged. (unfortunately for him, it was the authorities, and NOT mistress Nancy. )
Well, Brewer’s Dictionary of Phrase and Fable offers:
Spread-eagle (To). To fly away like a spread-eagle; to beat. (Sporting term.)
“You’ll spread-eagle all the [other] cattle in a brace of shakes.”- Ouida: Under Two Flags, chap. ix.
The “Spread Eagle,” meaning an eagle represented with its wings and talons outstretched comes to us from heraldry, and is so described as our National Emblem. The jump to “spread-eagle” as a pose is easy, and is left as an exercise for the reader.