you are right in our local language the two words that are written the same in English actually have different meanings in our country (indo language : berhemat dan menabung). I also have quite a hard time conveying the meaning of the other word because I haven't found a suitable vocabulary in English.
well you are right that they probably do have the same mindset as you describe. they work for money. but money also works for them. the point is they can take advantage of both sides. And it's no wonder the rich are getting richer. because their mindset is sometimes more advanced than ordinary people. and the way they manage their finances is also very neatly organized and planned. so that evaluation and observation becomes easier to do.
In many cases, the result is achieved by the one who acts, sometimes a quick but not the most correct decision may be better than a slow and balanced decision. But wealth can be achieved by people with different characters and types of thinking. The most common trait among the rich is thrift, they all know the value of money and know how to value their time, they work hard to create cash flow, so you are right, they do everything to make money work for them.