Not sure what you mean by 'register the block height of a transaction', but the likelihood of a block being orphaned becomes exponentially more improbable the deeper it is in the chain.
I mean I was doing some calculation on the following transaction taking it as a reference.
https://blockchain.info/rawtx/114f37f3e30c7d9b5d5339e9a1652f2f065d0413fe34ee1121de0f21a50aaed3As far as I can remember, the block height I registered for this transaction was 273824. But now it seems, its block height is 266559 !!! So, I am confused, if it is my mistake or block height can change with time !!!
You can not specify what height your transaction is in the block. It is up to the miner to include your transaction, or not.