Are bitcoins (or fractions of bitcoins) labeled with IDs?

Are bitcoins (or fractions of bitcoins) labeled with IDs?

Many sources describe the double spending problem by saying something like "people cannot be able to spend the same bitcoin twice." [1][2][3] So are bitcoins, or the smallest allowable bitcoin fractions, distinctly labeled entities? My understanding was that they are not, and that the blockchain only records information on the amount of bitcoins each user has.

Also, if I am correct, what would double spending actually look like in the bitcoin system? If I send a transaction to one miner saying "I pay user X 1 bitcoin" and another transaction to a different miner saying "I pay user Y 1 bitcoin", then it seems like I haven't double-spent, but rather just paid two bitcoins in total, one to each user. Right?

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