This is the critical part your requirement, since it's important to get it right for emails then Full Name would be the correct option. If you needed the name for business or legal reasons then you'd want to go the other way.
If someone's name is super long "Christopher Alexander Worthington" and I store that as full name.
Or someone enters "S. Smith" as the full name.
I could either put the whole name in emails "Christopher Alexander Worthington requests signature" or "S. Smith requests signature".
But in cases where I just want first name, I have to "guess" and hope that split [0] display makes sense.
"S. is inviting you to ..."
I feel like I'm better off sticking to first/middle/last and having people conform to that, given the limitations.
If I need to do invoicing I can just construct the name. For less formal things, I can just pick the first name which is required anyway (and last name).
You could also have some examples of how their entry will be used which follow your currently defined business logic, so
“Full Name:” S. Smith
Example: S. shared this with you
S. wants to be your friend
Let the users get immediate feedback about their choice