I liked the way that one of our wooden boundary pegs I found was nailed completely above ground to the fencepost that replaced it
The two words monument and corner can be synonyms. However, when the physical representation of the corner is moved, then it no longer occupies the corner.
An original monument creates the corner (with exceptions of course) but when it's disturbed like Bruce's example the corner doesn't move with it. Now the monument is at best a reference monument. Scribe an X on the transformer and show the B&D from the now reference monument to the corner. Or pull the dang thing and set a WC on the property line.
Find a crumbling set stone and replace it with a brass cap, bury the stone alongside and now the original stone monument is a memorial.
I would call Jim Cox's example a memorial monument, maybe it could be made into a reference, doesn't sound like it.
When you figure out what type of object you're dealing with, it's very helpful to complete a survey. Clearly unless you have a time machine much of what we do is conjecture and evaluation of evidence gleaned from our experience.
But using correct terminology is very important.
monument, corner, reference, witness, memorial, accessory, existing, existent, obliterated, lost, disturbed.
Figure out what the correct term for the object you're looking at and you're a long way to finishing the survey.
how can one say whether or not a found monument at one of its corners is an 'original' one that should be accepted if there's no record of when or by whom it was set?
This is when you ask the old surveyor what he usually finds in that plat. Honestly, I moved to a new county, and without a long time surveyor on staff here, it would be very difficult to make some of those decisions.