My teach pendant needs to be replaced. How do I identify the exact replacement. I know I have a UR10e arm. Is that all the information I need. I cannot find a model# or serial# on the teach pendant itself:
does it have the 3 position switch on the back?
It has a single button switch on the back. I believe it is for “free drive” of the arm.
I am considering replacing the cracked screen myself and am looking for a part# for it.
Well. If you were to order a new teach pendant, which is the “correct” way to do this, you would just need to show your local distributor those pictures and provide the serial number of the robot. 
There are just two teach pendant types for the e-series (as WattsUp was pointing out); without 3-position switch and with.
The teach pendant is made by UR, and they do not provide any information regarding the internals. So if you wanted to replace the screen only yourself, you would have to identify the part yourself by taking the teach pendant apart.
This is a standard teach pendant, it is the default for all e-Series (except UR20 and UR30).
As @efn has said, contact your distributor with the robot number and they will be able to provide you a solution.
Last time I checked, a new pendant was around $2500.
I believe we did an exchange for $1500. They asked for a photo of the unit, so it apparently depends on condition.
We have a couple pendants with damaged frames around the screen - got dropped before we started putting protectors on them - but they still work ok and the protectors cover most of the damage.
The vendor we purchased most of our robots from now sends them standard with screen protectors and Pendant Armor (looks like what you have on yours) protectors on the whole unit.
We did find that the Pendant Armor protectors don’t like the rust preventive oil that we use.
All our parts are covered in it, so at least in one of our inspection areas, the operators’ hands transfer it to the pendant protector.
This causes the Pendant Armor to swell so much that after less than a year the pendant protector was basically just flopping around.
We’ve switched those units to a pendant protector made by DRIM. Different style without the foam that Pendant Armor uses, but has so far held up to the oil.
Thanks for the information! It sounds like a bit of a risk to attempt a DIY repair.