I’m not familiar with any savings accounts that dont permit you to take money out whenever. You might mean CDs, which will, as per the agreement, assess a fee against you if you take it out early. But you still have title to the cash whenever you want. It’s yours. And if a fee happens, the fee is still money that is now owned by the bank instead.
Down the line you suggested depositing money creates money because now you and the bank have a dollar. That’s not true, because the bank doesn’t have title to the dollar.
If I ask you to hold my drink, you’re not allowed to drink it. Banks have rights to do certain things with your deposits, but it’s not theirs. They can’t decide to pay salaries and deduct the amount from the deposits.
It is part of M0. It's more "money" than deposits (M1) or savings accounts (M2). In a discussion about money supply it definitely counts as money. And they have it. Yes, they need to keep enough reserves but apart from that it's their money to use as they please (lend, invest, spend, pay dividends...).
The deposit is their liability (obviously they can't spend it). The dollar is their asset. When you deposit the money into a bank that dollar is not yours anymore, the bank is not keeping it in custody for you.
Down the line you suggested depositing money creates money because now you and the bank have a dollar. That’s not true, because the bank doesn’t have title to the dollar.