Way here is used in the sense of “path”, literally, or method, figuratively; and VERB our way is a common idiom meaning to traverse that path by VERBing.
For instance, one of the most chilling moments in drama is Goneril’s line in King Lear, after she and her husband have put out Gloucester’s eyes:
Go thrust him out at gates and let him smell his way to Dover.
Almost any word can serve as VERB; here are a few examples from Google:
How German Companies Bribed Their Way to Greek Deals
Wynonna and partner Tony Dovolani—who was Sara Evans' partner years ago—cha-cha-cha’ed their way to an 18 out of 30. (The cha-cha-cha is a Latin dance)
The team that could build the “coolest” machine out of a pile of scrap metal and spare parts would receive a $10,000 prize. In the meantime, the festival-goers could look on as the hackers DIY-ed their way to steampunk Rube Goldberg bliss. (DIY is do-it-yourself)
‘Did the Patriots cheat their way to another Superbowl? … ‘No, they Tom Brady-ed their way to another one.’ (The New England Patriots are a football team, and Tom Brady is their quarterback)
Ron is saying "To get across the room we are going to have to play chess."
As was pointed out on that other question you linked, this is a surprisingly tricky sentence!
Let's slowly build it up.
It will take someone else.
What is "it"? Stopping Voldemort again. Implied but never directly said. As for "take", we could say "need" instead, just to be slightly clearer.
[Stopping Voldemort again] will [need] someone else.
This isn't "someone else" as in "we need a different person", but as in "we need other people prepared to do what you did, Harry". When will they be needed? "Next time" (that is, the next time that Voldemort tries to return).
[Stopping Voldemort again] will [need] someone else who is prepared to [do the same as Harry] next time.
Now, Dumbledore is saying that this someone else doesn't have to do much. Hence, it will "merely" (or "only") take someone else. This may sound like it's demeaning Harry's efforts, but it's meant to be reassuring Harry: standing up to Voldemort is not very difficult or unlikely after all, and it doesn't take someone extra-special to do it.
[Stopping Voldemort again] will merely [need] someone else who is prepared to [do the same as Harry] next time.
What, exactly, did Harry do that Dumbledore says they need other people to do (or be prepared to do)? "Fight a losing battle"--or what seems like a losing battle. (Dumbledore omits the word "like", but I'll leave it in for this one example.)
[Stopping Voldemort again] will merely [need] someone else who is prepared to fight a losing battle next time.
[Stopping Voldemort again] will merely [need] someone else who is prepared to fight what seems [like] a losing battle next time.
And now we just replace the bracketed bits with the different wording Dumbledore uses, and we have the sentence (okay, part of a sentence) that you bolded.
It will merely take someone else who is prepared to fight what seems a losing battle next time.
Best Answer
According to the Cambridge Dictionary, the expression "better you than me" has this definition:
This is a mainly US expression: the UK equivalent is "rather your than me". This fits with the first definition that you quote, but not the second. I agree with the Cambridge Dictionary and your first definition: I think that the second definition is wrong.
A better expression for the second definition would be "I'm alright Jack"