IANAL, but. As you note, when you open a new account, they give you temporary checks that are usually blank in the upper left. I've used such checks and the bank has honored them. Therefore, I conclude that there must not be any legal requirement for anything to appear there, nor does the bank require it.
Businesses are often reluctant to accept such temporary checks, for the obvious reason that anyone could go to the bank, open an account with $10, write checks for thousands of dollars, and disappear. At least if they've waited long enough to get the permanent checks in, there's some reason to believe that they plan to stick around.
In any case, it's not clear what you are trying to accomplish. You want to hand-write either your business name or your personal name depending on whether the check is for personal or business purposes? I don't see what that gains. You could always use a personal check for business purposes. If you're afraid someone will say, "Hey, that doesn't look very professional, what kind of fly-by-night company is this that uses personal checks?", surely a hand-written company name would look even less professional. Why not just open a business account and have your personal checks printed with your personal name and your business checks with your business name? I don't know where you live, but I have a business account on which I pay zero fees. The only cost is getting checks printed. There's the small hassle of having to make one trip to the bank to open the account. Well, the biggest hassle I have is that the bank won't let me transfer money between my personal and business accounts over the Internet, so I have to either go to the bank to move money back and forth, or I have to write a check from one account to the other and deposit through an ATM.