You have already answered your question. The basic double spending attack in question is just that: The attacker initiates a Bitcoin transaction and claims a benefit from it. Then the attacker does the same thing again, potentially broadcasting the new, conflicting transaction to a larger number of nodes in the Bitcoin network, in an attempt to have the new transaction confirm instead of the first one. If that succeeds depends on which kind of miner, one having seen the initial or the later transaction first, manages to mine the next new block (with a few extra caveats, like that miner actually choosing to include said transaction in his newly mined block).

Computer scientists call this general phenomenon where the overall outcome from concurrent processes depends on what happens first a race condition. The race condition I just described can continue if nodes continue mining on different subchains where one contains the first transaction and the other subchain contains the other transaction from the double spending attack.

There are also more complicated forms of double spending attacks, but all of them have a race condition to some extent, and hence I would call all of them eligible for the description "Race Double Spend." You may also be interested in one, the Finney Attack, that makes the race condition almost disappear through premining, such that the attacker has an almost unbeatable head start if the recipient (e.g. the merchant) instantly accepts an unconfirmed transaction.