What does it mean that "am" is the first person singular present indicative of "be"? What is the definition of "am"?

Let's try to break this down:

The basic form of a verb is usually expressed as an "infinitive" e.g. "to do", "to make", "to run", and so on.

When used in a sentence a verb is modified to indicate when the "action" took place and the subject that "performed the action".

"when" an action took place has three simple forms (with multiple sub-forms):
#color(white)("X")[("past"), ("present"), ("future")]#

"subject" has two components:
#color(white)("XXX")#"person": relation to the speaker;
#color(white)("XXX")#"number": singular or plural.

In English, the subject may be:
#color(white)("X")[(color(white)"XXX",,underline(color(black)("first person")),underline(color(black)("second person")),underline(color(black)("third person"))), (color(black)("singular")," | ","I","you (thou)", "he, she, it"), (color(black)("plural")," | ","we","you","they") ]#

to be is an "irregular verb"
This means that it does not follow the most common patterns for forming the verb agreement with "when" and "subject".

#[(underline(color(black)("to be")),,underline(color(black)("usage"))), (color(black)("past")," | ",), (color(black)("- - first")," | ","I was"), (color(black)("- - second")," | ","you were"), (color(black)("- - third")," | ","it was"), (color(black)("present")," | ",), (color(black)("- - first")," | ","I am"), (color(black)("- - second")," | ","you are"), (color(black)("- - third")," | ","it is"), (color(black)("future")," | ",), (color(black)("- - first")," | ","I will be"), (color(black)("- - second")," | ","you will be"), (color(black)("- - third")," | ","it will be") ]#

As can be seen from the above, if we wish to indicate (use the indicative) of the verb to be to express the present state for the first person we should use the version: am