There are a lot of posts about this topic here on CC. But overall, bakers will tell you that once the cake is handed to the customer it is their responsibility.
However, I think it is your responsability to inform them of the precautions to take (transport on flat surface, etc.) and to give them a cold cake.
Personally, I do not agree to give cakes directly to the customer with more than one tier. They have to pay for delivery.
So really it is up to you to determine your policy on cake deliveries.
As long as you know it was solidly constructed and ready to transport safely then it's the customer's responsibility to get it where it's going safely.
Having said that, if the cake wasn't refrigerated before delivery there's a high probability that it will collapse if someone who isn't used to moving cakes is moving it. Was the one you gave the customer cold or room temperature? Whenever anyone on here says that a cake collapsed 9 times out of 10 it was room temp, and cake is just softer and more prone to damage then.
If a customer wants to take a cake you need to make sure it's built more solidly than you usually would, since you have to take their lack of delivery experience into account. Make sure the cake is cold, boxed up safely, and that you put it in the car. They shouldn't have to be responsible for it until they drive off. If you hand a cake to a client and it's room temp, boxed in a flimsy bakery box, and they put it on the sloping front seat of a car, damage is going to happen.
Not saying that any of this is what happened in your case. I'm just making the point that it's one thing to say the customer is responsible for the cake once they take it, but another to actually do enough on our end to make sure that they'll be able to move it without incident.
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