Where is the center of a rail monument? Is it the center of the web or the head? One found a quarter mile away has a cross in the head.
Is this the type you are seeing? I occasionally find these with a "+" in the center of the web.

Yes, like that. Guessing I'll shoot both the web and center and what fits better.
I mean the web and the center of the head.
Jerry's picture agrees with where I would take my shot. HOWEVER, the guy who placed it there had no expectation of it being judged so precisely. In fact, since the entire rail is the monument, anywhere you choose to use for your purposes is fine. The next surveyor will come along to find the same rail and take a shot wherever he determines he prefers, despite it now leaning 0.04' from where you found it. His numbers will disagree with yours, but, both of you were just as accurate as the other because YOU FOUND THE MONUMENT. That is, you found the corner, because the monument is the corner. The corner is not a perfect set of coordinates.
The center of the rail on the outside.

That is interesting, but I don't believe this monument was set for the purpose of just locating the R.O.W. line. It is on the R.O.W. line at its intersection with a street line and its base is at about a 45° angle to it.
As absolutely rational and reasonable you are I'm not sure that is going to fly straight with everyone. I mean aren't you going to narrow down the point to measure from to under a hundredth and then massage it with all the other info with a least squares adjustment so you may report an error ellipse of say 0.02 x 0.04 for that rail monument. Geez, if you don't the next guy might be unable to find the monument.
OK, I'm being a little sarcastic here, I'd probably look for a mark to measure from, I just wouldn't get all bent out of shape if my measurement didn't match up to the record exactly. Logically the center of a rail would be the center of the web from both directions to be. Circles and squares are easy. Stones, well I just sort of estimate and pick my spot. Sometimes you need to straighten things up a bit. There's just some slop in there, has been from the beginning.
That diagram is one I drew up using information from a PLS session given by Charlie Tucker and Gary Gable. I've given it out to party chiefs when they are looking for railroad ROWs. I've also had some turned like you describe. You never know if they were originally placed that way or were disturbed. Makes it harder.
I'd accept that if is was the RR that set it. Seems they tried to keep everything inside the ROW including their fence. On RR ROW's I've worked the fences are usually narrower by a foot or two than than the ROW width. I do not, however, have extensive experience with RR ROW's, just the few where a boundary I'm doing intersects.
Charlie Tucker is doing a one day session at the upcoming UCLS (Utah) conference later this month. I'll be there!
Heh I shot that rail and it is off 0.01' north and 0.02' east of the "true" point.
Mighty:
When used for R/W markers for highways, what your diagram shows is where the R/W line is according to the big boys that I've talked with.
My understanding is same as MightyMoe. I'm betting any marks like the one J Penry showed are often the result of retracing surveyors marking where they thought they should shoot the rail end. I highly doubt the railroad marked the rail ends since the standard was for the positioning to be as Mighty Moe has shown.
Rail Monuments Seldom Are Plumb Enough...
to justify holding such a mark. They generally are tall enough that nature has her way with them. It is doubtful that the more precise RR surveyor was carrying them around to set as monuments. Most often they were set much later by maintenance crews to mark where they think the wooden stakes once were. Short pieces of rail were useless scrap and could have been utilized by any surveyor in the area. A few years ago I surveyed a parcel that was bisected by a trolley line. I found trolley rail used thousands of feet away as markers.
Paul in PA
Yes, Agree with Moe

One day probably isn't enough. I groaned when I saw what that year's sessions were to be about, but I was wrong. It was one of if not the best one I've been to. A lot of information and it was interesting. Be prepared to have some assumptions shot down.
http://www.aocweb.org/surveyors/oregon_railroad_maps.ht m"> http://http://www.aocweb.org/surveyors/oregon_railroad_maps.htm
This is the only guidance I have ever seen on the subject in this area.
I am not an expert in railroad retracement, but I would guess the standards and expectations vary by region.
Maybe a r/w specialist for the railroad you are retracing could offer some insight?
When I worked on a DOT construction survey crew, we set highway right of way rails per MightyMoe’s diagram. We used a steel tape, transit and the right of way acquisition plat to establish the station and offset to right of way break points.
When the resident construction engineer was looking for busy work for his inspectors during the winter break in construction, he would send a couple of them out with a pickup load of rails, a couple of 16 pound sledges, a cloth tape, a right-angle prism and a set of plans that showed right of way breaks plus or minus.
caveat agrimensor
Moe is correct according to the BNSF Railway.