Question about staking - what does it mean for non blockchain protocols to have staking? Is it the same as PoS?

Long time fan, first time caller

I have a question that's been bothering me for a while. In regards to PoS, I can follow the model through the whole process... when we are talking about Ethereum (or any other PoS-secured L1).

To keep it short and sweet, my understanding of PoS comes from the analogy that PoW is essentially a footrace that, although has some elements of randomness, rewards the fastest racers (highest hash rates) with winner take all rewards. For PoS, we skip that race and instead procedurally distribute the awards based on participation (rewards distributed based on stake). This is both more democratic as anyone with capital in the system can benefit from securing it (as opposed to just miners), and collapses the energy cost of the whole endeavor. I know that's pretty loose but let me know if I'm way off.

My question is about projects that have "staking" but aren't blockchain networks themselves. The whole point of that game I just rambled on about there was to distributing the rewards for maintaining the system. Very specifically, it’s about allocating new supply at the leading edge of the blockchain to the consensus protocol. But if you aren’t an actual blockchain that has new supply/inflationary powers, then what does it mean to stake?

For example, right now CRO and OHM are some of the popular projects that feature staking. Is this the same staking at play in the PoS that (will be) secures Ethereum? Are Crypto.com and Olympus DAO independent blockchains? My other thought is that in this context, staking might just mean giving a protocol your liquidity in exchange for some sort of rewards.

If someone could help me understand where I’m confusing myself and/or point me in the direction of any resources you think might be helpful I’d very much appreciate it!