As one would expect, reflexives in the binding-03 grammar tool need a c-commanding antecedent that agrees with it in person and number, when it is marked for these features. Gender is not marked so the word 'him/herself' is both masculine and feminine.
The most important feature of the grammar in binding-03 is the difference between the forms 'him/herself', 'myself' and 'yourself', on the one hand, and the form 'self', on the other. The first forms are often called "complex reflexives," and the second "simple reflexives." In this grammar tool, the complex reflexives must be locally bound, that is within the domain of the first c-commanding subject. The simplex reflexive, on the other hand, cannot be locally bound. Instead, it must be bound by the subject of a higher clause. However, it can only be used if its own clause is untensed. The one exception to this rule for 'self' is that it is obligatory with an inherently reflexive verb ('behave' in the grammar tool). Surprisingly, this apparently schizophrenic behavior of the 'self' form is actually found in several natural languages, among them Dutch, Norwegian, as described in the text, and Chinese also show similarities to the grammar of the tool.
Here are some more details that you might have noticed:
If the reflexive is inside a DP in a postnominal PP, the behavior of the reflexive depends on whether the DP has a subject. If the DP has a subject, then the reflexive must be complex and its antecedent must be the subject of the DP. If the DP lacks a subject, then the reflexive maybe be either complex or simple. If it is complex, its antecedent will be the subject of its clause. If it is simple, its antecedent will be the subject of the higher clause.
The complex forms can occur in the subject position of a tensed clause and be bound by the higher subject. The simple form cannot. This suggests that the complex forms can bear nominative case but the simple form cannot.
The antecedent of a complex reflexive can be any c-commanding DP, not necessarily a clausal subject, but the antecedent of a simple reflexive is apparently always the subject of a higher clause. There is, however, not enough data in the tool to be sure that the antecedents of simple reflexives must be subjects.
If a sentence contains three clauses, a matrix clause, a subordinate clause, and a second subordinate clause embedded in the first subordinate clause, and if the subordinate clauses are both non-finite, a simple reflexive in the lowest subordinate clause is bound by the subject of the next higher clause, not by the matrix subject.
These remarks do not exhaust the behavior of the tool, but are the main points of interest.
Here are some trees generated by the grammar tool. As defined in the tool, the red outline boxes indicate which DPs are possible binders for the reflexive in blue.