The first sentence is grammatically correct. "Hirobumi Ito was a moderate who wanted to take economic gains first." In this sentence, Hirobumi is already established as a moderate, and also wanted to take economic gains first. Even if he did NOT want to take economic gains first, the structure of this sentence implies that he would still be a moderate, whether or not he wanted to take economic gains first. You might be comparing him, for instance, to another moderate who did NOT want to take economic gains first. In my opinion, the meaning of the second sentence is unclear. If the idea of the sentence is that taking economic gains first is the thing that makes Hirobumi Ito a moderate, I would suggest two alternate formations: "Hirobumi Ito was moderate, to take economic gains first." (Eliminate "a." This version emphasizes that Hirobumi was moderate in this specific action (taking economic gains first), but maybe he is not moderate all the time.) "Hirobumi Ito was a moderate, because he wanted to take economic gains first." (This version emphasizes the opinion of the writer of the sentence, who associates moderation with taking economic gains first. It is as if you are proving your assertion that Ito was, in general, a moderate, by pointing out his policy of taking economic gains first.)

No. The second is at best unclear, and at worst ungrammatical. It is missing a verb. "To take economic gains first" is a prepositional phrase that modifies "wanted." It says what he wanted. It is like saying "I ice cream" instead of "I want ice cream." You would never say the former, because it has no verb. The 2nd is grammatical if it means "Ito took economic gains first. That shows he was a moderate." (Similar to "You are a bad person to steal from a church.") But that does not mean the same thing as the first.

Both are grammatically correct, but mean slightly different things. One says he wanted to .... The other said that he did, or does ... . In this sentence they are virtually interchangeable.

This is a question to answer. He was the one to do it. He was the expert to answer. He was an expert to answer the question. He was the type of moderate man to take up a challenge. He was a moderate to take [whatever you want]. . The answers below are reacting to the slightly unusual structure of your second sentence. In fact there's nothing wrong with it. It's just an interesting structure, less commonly used. . I suspect part of the problem with recognising it is that the first sentence biases people on what they are looking for, It biases them against being open to the second structure. . "to take economic gains first" is adjectival on Ito. If you like, qualifying the type of moderate he was. It's fine. With the context given by the first sentence, some may misinterpret it and be confused. . This was a question which confuses some. This is a question to confuse some first. Enjoy.