While yes, there is a Preference for Assigning Defaults, what you just described is Variability, not Default.
You stated this: "Various accounts are deposited to one of the two"
In QB, it's Transactions. Not "accounts." For instance, Payments against Invoices or sales receipts for Services or Dues, vs Sales Receipts for donations and Grants. And instead of micro-managing by Accounts, you would be using Class tracking. Example: My Pledge to you, put on an invoice, includes Dues + donation. Class for Dues = Admin or Operation. Class for the donation = your program or purpose and the Donation item is a Restricted income activity. That entire payment I make gets deposited to Checking. You run Balance Sheet by Class to see Admin funds, Programs/Purposes, all as Columns. Your Income and Expense, P&L By Class, is also broken out like that.
I recommend the book Running QB for Not For Profits, by Kathy Ivens. And you need to be using Premier. Pro does not manage Bal Sheet reporting by Class.
There are two functions for default Deposit. Payments preferences > Company Preferences, always use Undeposited Funds by default. That means sales receipts and receive payment show No Dropdown as "Deposit To:"; you already told them where to be parked. Then, you select which transactions are going into which Bank account by using Banking > Make Deposit.
In Checking Preferences, you control Open Deposit to this Specific Bank account, as Default or Initially. That means, for your example, when you use Banking menu > Make Deposit, you select from the UF transactions, then the next screen, which is the Actual Deposit worksheet, has the same specific bank account top left, every time it opens. Example: 9 times out of 10, I want Checking, not Money Market. So, I set this function.
