I saw myself in the mirror.
I solved the problem myself.
I was all by myself in the room.
I brought an umbrella for myself.
I understand different usages of reflexive pronouns. However it confuses me when I see a structure with a reflexive pronoun like the one below.
'We also place a little more emphasis on the nature of the action itself'.
I also found some other instances of this structure.
'I saw the man himself.'
'She got to dance with the prince himself.'
As I can see in these three sentences, the reflexive pronouns don't match the subjects of their sentences like the previous ones.
I would appreciate if you could help me understand the reasons behind this rule, and when I should use this rule.