Blocks propagate ‘almost’ always within time to all other nodes before the next block gets mined in approximately in 10 minutes.
The 6 blocks waiting time is just for being sure that your transaction is on the blockchain forever.
Because the more blocks get mined since your transaction is in a block the more difficult it gets for a miner to mine a larger chain within the time a new block gets mined again.
*Every new mined block on top of the block where your transaction is in, means a new confirmation)
So if your transaction has 6 confirmations, it means that 5 new blocks are mined after the block where your transaction is in.
If 2 miners mine a new block at the same blockheight around the same time, you basically get 2 different blockchains propagating to the network. Each blockchain will have a different last block on that blockheight. These are 2 valid blockchains according to the bitcoin consensus rules.
But eventually, 1 of the 2 different versions of truth will get dropped. Because in Bitcoin, we have a rule that nodes with the longest blockchain (with more proof of work) will be the accepted blockchain. So once another miner successfully mines a new block on 1 of the 2 different blockchains, It will become a longer chain with 1 extra block than the other competing blockchain, with a different block. That block will become a stale block and all transactions that aren’t already included in that other accepted chain, will go back to the mempool.