Answer: The correct analysis is the first one: it contains two compound sentences and one complex sentence.
Explanation: A compound sentence has two related independent clauses that are linked by a conjunction like AND, BUT, SO, YET, OR, etc. The sentences "all the students signed a petition, but they still needed the faculty’s permission to start the club" and "the students came to the next faculty meeting, and they presented a solid argument" have the coordinating conjunctions BUT and AND. So they are compound sentences.
A complex sentence has an independent clause and a dependent one. The ideas are usually linked by subordinating conjunctions like BECAUSE, AFTER, AS, BEFORE, WHILE, etc. In the sentence "all of their hard work paid off because the students won their campaign" the subordinating conjunction BECAUSE is used to join the dependent and independent clauses.