Can you say exactly why the transaction would be valid only on that branch?

A fork occurs. There is demand for coins (and mining occurring) on both chains. So, we have two surviving block chains (for whatever period of time).
You will then have 1 XTcoin and 1 traditional bitcoin.
You can then taint one of your coins with coins which only exist on one of the forks. Let's say you get a few satoshis from block reward which occurred in a XT 8mb block (thus is not valid and does not exist on the 1mb chain). You can combine those satoshis with your 1 XTcoin by sending them to a new address under your control. This means your XTcoins can be safely moved separately from your traditional bitcoins.

If you don't taint your coins first, any transaction you make will be valid on both chains, so if you send your coins to an address which is not under your control, the holder of the private keys for that address will be able to get both your XTcoins and traditional bitcoins.
There is no risk for you, until you make the decision to taint your coins on one chain, and then exchange them for something of value. You will hold coins on BOTH chains until you take these actions. Even the tainting itself does not incur any risk for you. It simply allows you to craft transactions on each chain which is no longer valid on the other chain.