Thought I'd chime in here,and mention something unique to 51's that is often overlooked.
51's had a grounding wire lead off of the pick-up,this wire was seated beneath the pick-up cover.
Spent some time with an original 51' many moons ago,and this is something that kinda stood out with me.
I often wondered why this feature was dropped from subsequent models,because it did change the function somewhat.
With the cover on/grounded,the single coil hum was toned down,interference was minimized-and the sound of the pick-up was smoothed out.
Guess you could say it was toned up

Maybe with the old 2 prong power cord style amps there might have been some unpleasant zaps? who knows?
Recently,a TB'r sent me a very nice pic illustrating this set-up.Although this particular 51' was a reissue,it was a Mark Kendrick,masterbuilt.
And it really blew my mind,that Kendrick,went to the nth' degree on this build.
Here is the pic.
There has been some good info posted here so far,but I'd like to also talk about the neck profile of these early P's a bit more.
Because to me this is one of the main distinguishing factors that set the early P's apart.
M.R.Ogle,provided a brilliant illustration in
post #27.
I currently own a Fender Custom Shop 51' P that was made circa 1991,this was the fortieth anniversary of the Precision,the custom shop made a special run of 51 basses to commemorate this milestone.These are pretty special basses,very similar to the originals.
I also own a 55' custom shop P,and both these basses have the same neck profile,as illustrated in Ogle's post.
To me this neck profile is pleasingly plump

and a big contributor to the early P's distinctive sound.
The more recent 51' P reissues by Fender,Japan,do not have this neck profile,they are close but no cigar
If you can check out either a custom shop 51' or 55' reissue you will be able to see and feel what I'm talking about
Hope this helps.