Definitely not old school. Great point. The nomenclature staking evolved within crypto from the PoS coin staking itself.
I am sure you individually know about it but just to add to the conversation, the second definition of staking evolved from the trend of yield farming on DeFi. There too, two terms have mainly evolved. Liquidity mining and Staking. There are pools and there are farms. The pools are generally for pooling liquidity where you put half of each asset (a stablecoin and an Alt) and get a share of trading fees. This is where Impermanent loss comes in.
If you are just Hodling, there is no IL. I think OP may even be researching IL because that is where the difference between Hodling and yield farming (staking) is most prominent.
There are two different styles though, you have to remember that as well. I do not know if they are really different or not that much, but staking ETH and staking Cake are different for example. You have to remember that.
So, if you are supporting proof of stake coins, then Cake or Uni or anything like that is not involved in that case. I love to "stake" stablecoins for example, usually that is not called staking but you get some returns from Binance for example (and other places) like yearly 5% to 10% for staking them. With that type of return, you could arrange your life accordingly and live on passive income.
You could also do staking with coins that could earn you while gain value and that is great too. Both your income and your coins value would go up in that case, what you stake as 100k today and gives you 10k a year, could worth 200k if the coin does x2 and earn you 20k because the % is still the same.