First, the present participle is the word that ends in ing. The present participle can be used as a noun, in which case it can be called a gerund. For example, "Running is my favorite activity."
Your two examples are not using the participles as nouns, so they are not gerunds. But also there is not much consensus on what they're called, either. You can say that they are object complements. Participial phrase and relative clause are a couple more general terms that apply. In this case, we have a reduced relative clause because "to be" is implied in each sentence: "I don't want you to be crying," for example. So, there's also the idea of a present continuous infinitive in reduced form.
So, the more important thing is not what you call them, but how they function in the sentence. In the case of your first sentence, there's more of an adjectival function: "you" is the object, and we're using "crying" to modify "you." When you look at it that way, it's clearly not a noun, since it doesn't function as a subject or object.