To summarize:
POW:
Selects proposal based on what node is making it.(selects potential proposal based on node who first solved POW)
Ideal:
Selects proposal based on best proposal for the current conditions
(Selects node based on best potential proposal)
It's not any first, it has to meet the difficulty too. And this is important because it also regulates the speed of blocks being mined. "First" is relatively easy to quantify.
"Best" on the other hand... who and how to decide what's best, in an unbiased way, plus to make sure it's not cheated and also making sure the blocks are mined at the correct speed? (The who part is meant to be a trap, since the only correct "who" is the nodes.)
"Best" can be quantified by basic quantifiable metrics like transaction speed, value spent, number of transactions, who spent transactions, where in the network they are spent from etc. just depends what the network is designed/optimized for
Regarding First solution and block difficulty they are separate things. Block difficulty is constantly changing so the system approaches on 10 min block times. This means blocks will get easier or harder to solve based on how long the last block took to mine. When the network is operating smoothly nodes who propagate a solution to the POW function first are usually accepted as that makes that solution the longest chain. This usually means the first solution is accepted unless malicious/illegitimate activity is detected.