You are partially correct. Customer statements only include transactions processed through Accounts receivable or custom control accounts to which customers are assigned. However, when a special account is used to reduce the balance due on a sales invoice, the debit for the sales invoice is reduced on the customer statement. This is true for both types of customer statements. So you will not see the deposit on a customer statement when it is received, but you will see its effect when it is used.
The way to think of this situation is that deposits credited to special accounts are held in trust. They are liabilities of your business and not yet income. They are not actually applied to the customer’s account until you take action to do so. Then, they reduce the receivable asset associated with the sales invoice. So if you want them to be immediately applied to the customer’s account and show up in customer statements, you need to credit them to Accounts receivable. The consequence of that approach is that, because all money is interchangeable, the deposit will be applied to any unpaid sales invoices.