From your title savings doesn't only mean when you are wealthy, as a matter of facts wealthy people doesn't think of savings they think of investment where to channel those money whenever they are wealthy because one thing i believe is that, rich people talks of what will generate money for them while poor people think of work to keep them earning. The rich people think of wealth creation what will keep them afloat whenever its too tough for them to scale through life. In summary think of what will keep generating money for you than savings because i know too well that those money might got finished whenever you stop working to keep filling your saving account.
I've never heard of rich people prioritizing savings before investment, and I've never heard them took savings seriously. As what I can see from those people, they only treat savings as an emergency cash or spare money just in case something happens to them in real life or even on their investments. In the end it is all for their investments again.
Our perspective of savings are different, just for example, me that came from a poor family, of course I'll save enough money for my investment or to buy important things or my priorities. I need to save money and calculate even the smallest cents so that it wont go to waste. However, rich people will just invest almost 70% of their money while the rest goes to savings.