Can carrying a small credit card balance actually help my credit score?

No, this is a persistent myth. You do not need to carry a balance, or pay interest, to build credit. Your credit score is helped by using cards and paying them off, not by leaving debt on them. Carrying a balance just costs you interest with zero scoring benefit.

What's true, and where the myth comes from, is that having a reported balance greater than zero (versus all cards showing zero) can score marginally higher, because it shows active use. But 'reported balance' isn't the same as 'unpaid, interest-bearing balance.' You can let a purchase report on the statement, then pay it in full before the due date, capturing the tiny benefit while paying no interest. Use your cards, pay statements in full, and never pay interest to chase points.