The check register is for our use. Most banks do not care if you use the ones they provide or others, as long as your method helps eliminate overdrafts. Nor do they care how many lines in those registers you use per entry. I've worked in various banks in various positions, including helping people find their mistake when they don't agree with bank balance, &/or are overdrawn.
Thing is checks are not only thing you record in them, for ease of reconciling your account (i do weekly online) we should record everything-- debit card purchases, ACH Payments (automatic payments you authorize), deposits as well checks asap. Years ago i started skipping a line between most entries for easier reading of them. In the check # spot i put codes for type of transaction: DC, ACH, O/L or the check # for the 2-3 i actually write a month.
When i make a payment on line i put my abbreviation of Payee name (you only need full one on the check) on one line and the confirmation/transaction # on the second, even tho many companies will email confirming payment.
Do whatever works for you, while keeping in mind that some form of recording transactions as you make them is best way to avoid problems especially since so many older folks are more prone to memory issues.