The farmer grows the carrot and sells it to a merchant. The farmer pays the tax on the carrot. The farmer charges the merchant the price of the carrot + the 10¢ tax.

That's wrong. You're assuming that there's perfect price inelasticity for the carrot. Or in layman's terms, that the merchant is completely insensitive to the price of the carrot, and will buy exactly the same number of carrots as before - and the farmer can pass 100% of the tax on to the merchant.

In the real world, that almost never happens - and certainly not for goods like carrots. In reality, the farmer will not be able to charge the price of the carrot + the 10 cent tax. They will have to charge less.

The same holds in the exchange between the merchant and the customer. The merchant will pay more for the carrots from the farmer (but not 10 cents more), and the customer will pay more for the carrots from the merchant (but not the full amount of the delta the merchant paid to the farmer).

In theory, the amount of tax borne by the customer can be anywhere from 0 cents to the full 10 cents, depending on how price sensitive everyone is. In real markets, the value is always somewhere in between.