Homer was a Greek poet; he probably lived around 700 B.C. Is that a compound or simple sentence?

Although this sentence does not use any of the “FANBOYS” coordinating conjunctions, it is nevertheless a compound sentence. This is because the semicolon can take the place of a coordinating conjunction.
In a typical compound sentence, there are two independent clauses which are linked by a coordinating conjunction. For example, Sentence #6 in the screenshot you attached is clearly a compound sentence. It uses the coordinating conjunction “and” to imply Achilles’ anger was related to the fact that he only had one real friend. Sentence #1, about which you are asking, does not have a coordinating conjunction.
Even so, it is still a compound sentence. Compound sentences (as you can see in the link below) can be constructed using a semicolon instead of a coordinating conjunction. In Sentence 1, we have two independent clauses that could each form a standalone sentence. These two sentences are combined with a semicolon, which implies that the two clauses are related in some way. Because a compound sentence can be made using two independent clauses and a semicolon, the sentence you have given us here is a compound sentence.