The signature on the one original gun I have is oriented the way you describe. You read it like you hold it if you are right handed, butt to the right, muzzle to the left.
I also have an original signed barrel (barrel only, no other parts). The barrel is made by Roop, and is signed by J (or T) Ford. On this one, the signature is the other way around, you read it with the muzzle to the right, breech to the left. Based on where the touch hole is, it came from a right handed gun.
First gun is likely 1820's-30's. The barrel may be more towards 1805.
Cheers,
Norm