White Paper "Ending Condition 0" vs the next instruction

Questions and clarification regarding the steps listed in the White Paper “Algorand - a secure and efficient distributed ledger” from 2019.

Is the difference between “Ending Condition” and a very similar instruction in “coin-fixed-to-1” is
that the user also sees the HEAD message in the “Ending Condition”? These instructions look nearly the same. How does the user know if it end game or coin-fixing game?

In section 4.2, the ending conditions 0 and 1, The paper talks about a step s’ which is different than step s. In each BBA protocol sub-step (e.g. Coin-Fixes-to-0, etc) the first instruction is to check for ending conditions 0 and 1 (same instructions as in coin-fixes-to-0). I thought I understood. I’m definitely missing something.

In a coin-genuinely-flipped step, user checks the ending conditions again and then it lists (paraphrase)

“If at any point he has received greater than or equal to number of signatures of valid messages of the form where coin b is 0, then he stops waiting and sets his own coin be to 0”.

How is this different than ending condition 0? The only difference I see is the difference between step s and step s’. If these represent the same thing, If they receive the majority of messages for 0, wouldn’t that be ending condition 0? And if that is true, then how does it get to the next instruction that I paraphrased above (which pretty much has the same thing with the exception of step being s).

Please help me understand and clarify the difference between these.

Thank you,