When you sign something other than your name, say “Mum” or “Dad”, the word takes on the same importance as the signature. However, the image is narrower in it’s focus.
Whereas your entire signature is how you see yourself before the world, your signature of “Mum” shows only how you see yourself to your children.
The same, of course, goes for any other title, say “Grandpa” or “Grandma”.
So if you have occasion to write either of these, look carefully at how you sign those titles in comparison to how you sign your actual name in general, and find out how differently you see yourself before these members of your family.
Of course, there is always the “what you see is what you get” signature, where it is just the real you, no image, that shows. In this case, one would expect to particular difference in the “Mom” etc title either.
However if there is a problem between you and the family member to whom you are writing this title, it will show in how you write it.
Naturally, all this applies to letters you receive from “titled” people as well as those you write.