Budget savings inflow to the category?
I currently keep my savings in one account. I setup direct deposit with the savings account from my paycheck. When my paycheck comes in, those funds are input as "To be budgeted".
I currently keep the available amount on my "rainy day fund" category the same as the balance in my savings account and any spare change I have.
Does it make sense to change the transaction from paycheck to savings to the rainy day fund category?
The answer is the money should go to To Be Budgeted. And you should change your mindset. The money in the "rainy day fund" category is not the money in your savings account. It is simply X% of the money you currently have within all your accounts on budget. The fact the category available is the same as the savings account balance is just by chance (or a lot of unnecessary busy work). Please read this
What I am saying is I don't have an emergency fund. I don't need any category called such since using YNAB. Since using YNAB, I give a purpose to my emergency dollars by type of emergencies: car replacement, electronics replacement, house repairs, income replacement, bikes, furniture, appliances, health related costs. Having those funds separated allows me to be sure I'll be able to deal with the loss of a car and a broken water heater at the same time for example.
But once I have those categories funded monthly at a satisfying rate, I don't need to save more for emergencies. So if I have more funds, it is for more fun saving goals: vacations, trips to France to see my family, home renovations, glider, family time etc.
What is more important is what is explained in the doc I linked in my first reply. If you are using YNAB, you do not save money by sending it to a separate account. You are saving money by budgeting it in some categories and not spending from those. The only reason to have a savings account with YNAB is to make free money (interests). That's why lots of people using YNAB have savings accounts, but they absolutely don't try to sync one category with the account balance.
I am not in the same situation because of the country I live in (Australia) and I use my accounts differently, but there are a lot of threads around were people explained what they do to maximise the amount of interests they earn. Typically, it boils down to figuring out how much you are likely to spend in a month and keep this amount plus a buffer in the checking account. Everything else goes in savings. Or some are using scheduled transactions and the running balance in YNAB to see what will be their lowest balance on their checking. They decide they don't want their checking to dip below X and move anything extra to savings.
You might think then you will need to constantly transfer back from savings if you have money for some bills in savings, but in fact, you don't because your checking keeps receiving income. And that new income can then "become" the money to pay those bills and the savings money then "becomes" money in your savings categories. This doesn't describe a real action, there is nothing to do for this to happen as money is fungible. Or in other words, the location and the purpose of your money are 2 different unrelated ways to see your total amount of money.
Unless you plan to spend that money on the next rainy day, probably not.
Most of us keep all sorts of money in our savings account(s), just to earn some interest on it. We also have "emergency" categories we fund regularly without keeping track of which bank account that money is in - the category's Available balance tells us how much we've saved.
What is your Rainy Day fund for? Job loss? Income Replacement category. Medical emergency? Insurance deductible category. Your cat needed to have all of her teeth pulled? (happened to me last November!) Emergency vet category, etc.
It's really helpful to know what you're saving for. I didn't have quite enough in my Emergency vet category in Nov. so I pulled what I needed from other categories and moved on...