What exactly is a bitcoin? So I understand how blockchains work[1]. I understand how proof-of-work works[2]. And I understand how they work together. I also understand (or at least I think I do :) how transactions work (A block can contain arbitrary data inside of it, so it can contain a message signed by A that he transfers 5 btc to B).

But where does the bitcoin itself fit in this picture?

Is it a recalculated number every time (so if A gives 1 btc to B, then if I want to confirm, I verify that A has 1 btc to give away by checking who sent him anything, and make sure that he has what to give away, etc. up until the miner's bounty)? If so, this would have to be done whenever a block is mined (if not, I (who have 0 btc in my "wallet") can certify that I give B 100 btc, mine a block, and include that transaction. And while the blockchain on its own would still work (meaning that the hashes work out and solve the proof-of-work), the accounting wouldn't). 

Am I making sense or am I totally off?

[1]. It's a linked list, but instead of linking-by-address, it's a link-by-hash.

[2]. Change a part of the Node so that the node's hash would start with 100 zeros, for example.
There is more to a blockchain than just a linked list of hashes and a proof of work. There's more to Bitcoin than just a linked list of hashes and a proof of work. What defines Bitcoin and its blockchain are the consensus rules. These are a set of rules that determine what constitutes a valid block and a valid transaction. The blockchain is a linked list of blocks which contain transactions that conform to the consensus rules. Every block in the blockchain must also conform to these consensus rules.

One of the consensus rules is that a transaction cannot create more value in its outputs than it spends in its inputs. It can create less value, and the difference is the transaction fee. But it cannot create more value as that would lead to infinite inflation and Bitcoin would not work as a monetary system. This rule is completely independent of the linked list of hashes and completely independent of the Proof of Work.

Furthermore, there is no such an object as a Bitcoin. What people refer to as Bitcoin are simply values associated with transaction outputs. There aren't actually Bitcoins; it is solely based around the transactions and their outputs.