How does "spanner" come to mean "a wrench"?
Solution 1:
"Wrench" as a tool does stem from "wrench", meaning to twist:
wrench — Old English wrencan "to twist," from Proto-Germanic *wrankjan, from PIE *wreng- "to turn", nasalized variant of *werg- "to turn", from root *wer- (3) "to turn, bend" (see versus).
"Spanner", on the other hand, has this history:
spanner — 1630s, a tool for winding the spring of a wheel-lock firearm, from German Spanner, from spannen (see span (v.)). Meaning "wrench" is from 1790.
span — Old English spannan "to join, link, clasp, fasten, bind, connect; stretch, span," from Proto-Germanic *spannan, from PIE root *(s)pen- "to draw, stretch, spin"
The "wrench" connection is most likely related to the connection between "span" and "spin".
Solution 2:
BrE or AmE, a wrench is a 'wrench' and a spanner is 'spanner.'
Good analysis on WP: [emphasis added]
In British English, spanner is the standard term. ... The term wrench is generally used for tools that turn non fastening devices (e.g. tap wrench and pipe wrench), or may be used for a monkey wrench - an adjustable spanner.
…
In American English, wrench is the standard term. In American English, spanner refers to a specialized wrench with a series of pins or tabs around the circumference. (These pins or tabs fit into the holes or notches cut into the object to be turned.) In American commerce, such a wrench may be called a spanner wrench to distinguish it from the British sense of spanner.
See various types of spanners and wrenches.
Solution 3:
The spanner for the wheel lock gun looked like a wind-up key for a clockwork toy, with the major difference that the socket that fitted over the projecting axle of the wheel that was to be wound to tighten the inner spring was about three sixteenths of an inch square, and much more sturdy than the key for a clockwork train. The crossbow “spanner” was actually called a cranequin, for which your perusal of https://www.youtube.com/watch?v=zjYQEyC4m10 would be far more informative than my attempt at an explanation. In the next video he uses a windlass to wind up a more powerful crossbow. I had never thought of a windlass in this context. I had thought it was but a naval thingy for winding “sheets” i.e. ropes. The word “spanner” appears to derive from the “span” of the loaded crossbow being enabled by a separate device.
The next mystery is how the word became attached to the device for tightening hexagonal nuts, called merely a wrench in New World technology. Rupe's answer above appears to fit the bill best -- "A spanner is a tool that has fixed parallel flat faces opposing each other so as to be able to grip an object of the appropriate size. Because it won't grip an object that's the wrong size, that fixed gap spanned by the faces defines the usefulness of the tool (hence the name)."
Solution 4:
You should study the German verb spannen to see the whole breadth of its uses:
One can spannen a rope so that it is tight
One can spannen a horse before a wagon/waggon
One can spannen a workpiece into a vice
A bridge can sich spannen over a river
And a spanner holds the nut of a screw like a vice.
Of course, there are a lot of other uses:
A novel can be spannend (thrilling and similar expressions).
Solution 5:
Loading a crossbow was physical stress for the person who had to do this. Strong crossbows could not even been loaded by hand. To do this, a tool namend Spanner (german) has been necessary. It was like a arm of lever. When Screws came in application, there where also a tool necessary to tighten them.
The word was probably transferred from the medieval.