As you said, oligarch implies power, and with power one typically controls resources in a way that wealth can be generated for selfish purposes. The strict definition is something like "ruling by few", but I've always thought of it as rule by the extremely wealthy since that's how it typically seems to play out.
Sure, they are pretty closely related in pratice. The Senatorial oligarchy of Rome were pretty much also the wealthy of Rome, and often those with military power.
Still, the term implies politics... not just money. I would also hazard that it implies hereditary power. You could argue that dynastic politicians are oligarch. I'm not really disputing, just pondering.
Ultimately, for both Russians and Americans, oligarch is a (usually derogatory) analogy. It's not a formal oligarchy. We could also say aristocrats, which is more appealing to me.
Also, there's an idea that all governments, or systems of power are oligarchies. No matter what the formal system of government, some group of powerful people wield a lot of power. Rich people, religious leaders, senators, backscene power brokers, generals...
