I think your textbook might be incorrect. Also, be aware that translating Sp. a to Eng. to might be misleading in this case.
The last example should use a just like the first and second. Employing a before the object is compulsory in a number of cases, most commonly when it's a person's proper name, but also when the object is
- the proper name of an animal;
- a collective noun that refers to a well-determined group of people;
- a common noun that refers to a concrete individual;
- a common noun, even if general, where the meaning of the verb includes physical or psychical affect;
and many others (sources: Diccionario Panhispánico de Dudas). So for example:
Dejé a Fido atado a un árbol. ("I left Fido tied to a tree.")
Echamos a toda la gente afuera. ("We sent all the people outside.")
Fui a ver al doctor. ("I went to see the doctor.")
Dicen que golpeó a una anciana. ("They say he hit an old lady.")
Regarding which verbs always require a before the object, you have no alternative but to learn them by heart.
Le presento mi familia, just like that, sounds really odd to me. It would work if there was another explicit object in the sentence (which would be the referent of le):
Le presenté mi familia a mi amiga extranjera.
"I introduced my family to my friend from abroad."
Things can get a bit muddled because of the particular semantics of presentar (which is somewhat weird because it has two objects but it works more or less the same if you exchange their place, since you cannot introduce A to B without introducing B to A). But in any case you have to use a with it.