Probably in this case, according to this: http://spanish.stackexchange.com/questions/2119/what-is-the-difference-between-personas-and-gente
"The people count on her" Makes me think the people depend on her in some way. "The people count with her" Makes me think she cares for their needs. Both sound pretty similar to me. Either could be a lame slogan in a political ad. "contar(conjugated) con" means "to count on" so its probably the better translation.
Okay even if "count with her" does make some sense, Duo should consider what it wants to teach its first timers. English speakers would find a translation of "count on her" far more usable and relevant than a random sentence put together just for the sake of translation - one most people won't be able to relate to.
The Subjunctive is not a tense but a mood. What does that mean? It is called a mood because it doesn't deal with factual reality but with opinions, feelings, suppositions, dreams and speculation. We use the Subjunctive to mentally and emotionally organize our world in terms of others. Subjective thoughts and ideas based on feelings and personal viewpoint more than on objective fact. This is what subjunctive means in the grammatical sense to the best of my knowledge.