Hi! In this case, "se" is a reflexive pronoun, because the action of the verb "olvidar" falls on the same person who execute the action. It's a reflexive sentence.
It can be written of two ways:
"Se me olvidó la cartera" or "Olvidé la cartera"
Check out this link and its examples: http://www.materialesdelengua.org/LENGUA/morfologia/se/se.htm
About the other sentences where you use the verb "olvidar", all have the same meaning.
Regards.