Reflexives – Understanding ‘Reflexive Only’ Verbs

It’s an intensive or emphatic pronoun.

Per Wikipedia:

Intensive pronouns, also known as emphatic pronouns, re-emphasize a noun or pronoun that has already been mentioned. English uses the same forms as the reflexive pronouns; for example: I did it myself (contrast reflexive use, I did it to myself).

Here are the three cases you have presented:

(1) I gave Susie a picture of herself.

(2) I went there by myself.

(3) Do it yourself.

I have come up with a new rule under which your three cases as well as the traditional rule are subsumed:

A personal pronoun must be in the form of a reflexive pronoun in order to refer back to another word when both the word and the personal pronoun are contained within a single clause.

I'll call this new rule "the single-clause rule". The single-clause rule surely incorporates the traditional "subject-object" rule, because a subject and an object are contained in a single clause.

Now, let's see if the single-clause rule explains the three cases above.

Regarding (1) and (2), herself and myself refer back to Susie and I, respectively. In (3), yourself refers back to the implied subject you. And these are all in a single clause.