Being rich is not a guarantee to be wealthy. However, it can give us the highest possibility to be wealthy someday because funds are already there, we just have to make use of that the most profitable way. Saving isn't enough, we need to invest in different potential assets and diversify, this is the only way to gain multiple sources of income. And once it happened, you are no longer just rich, you're already wealthy enough because sources of funds have become unlimited.
You are right, and there is one more thing that must be noted, and it is that how can we identify someone as being rich? Or how do we know someone is rich? It is mostly by the lifestyle they have, right? Like having a big house, luxurious cars or a car, wearing designer cloths and other wearables costing a lot of money, etc. Now, we also need to understand that there is a very big difference between someone who is rich and someone who is wealthy, because when you are rich, you are basically just earning good money, but most of your money is just going towards the luxuries you have created and to maintain it, which often leaves you with nothing much that you can utilize to make investments or build assets for your future.
Now, the only thing that can create the possibility of someone rich potentially becoming wealthy over time is for them to use their income wisely. If you are earning good, don't spend all your money on creating luxury in your life, because it's not going to be permanent because as soon as you lose your job or your income, you won't be able to maintain the same lifestyle, and you will slowly lose everything, but if you spend your money in the right way and on the right things, you can have those luxuries for your whole life once you become wealthy.