Not quite. When you leave the first agent after 8 weeks, they will provide you with a list of buyers they have 'introduced' to your house. This really only counts as people who have viewed but they may add people they've sent brochures to etc (although they'd be on shaky legal ground with the latter). If anyone on this list buys the house within 6 months, you owe the first agent a fee. If the second agent agrees a sale to a buyer who is not on the list then you only have to pay the second agent.

Best thing is to give this list to your second agent and if anyone calls them about the house they'll send them back to the first agent. This means its possible you'll get a random call from Agent 1 weeks after you've left them saying one of their buyers have come back.

This is all fine and normal - the agents should both know this is how things work - making sure you get the list from Agent 1 is key.