What is the difference between an ACH payment and an echeck?

This is not true. I was paid by a customer via echeck and did not initiate the transaction. I just spoke with a representative in merchant services and they said an echeck and ACH are essentially the same thing. Echeck/ACH is any kind of electronic bank transfer including wire transfers. 

In my experience, the practical difference between ACH and eCheck transactions is that ACH transactions are typically approved or denied virtually instantly. 

With ACH payments through other merchant service providers, if the checking account doesn't have sufficient funds, the transaction is declined, and the merchant can require another form of payment while the customer is standing at the register.

When I use Intuit's eCheck transactions, it may be several days before the transaction is returned for non-sufficient funds.  You should treat eCheck transactions with the same level of scrutiny as you would a paper check.

Wire transfers are something else entirely and don't directly involve the merchant at all. Also called SWIFT (Society for Worldwide Interbank Financial Telecommunications) transfers, they are direct bank-to-bank transactions.  In the US, they are usually only used to send money internationally.