Tea cosy double rib question

I have two patterns for basic tea cosies in double rib.
They both say to cast on an odd number of stitches and to do K2, P2 to last stitch and then K1.
My question is this: when I turn the work round to work back I won’t be doing K2 P2 right under the previous ones because of that odd K1 at the end.
How does this work?

You’re right - you’d start the next row with P1, then go into k2, P2 across.

No, wait! Double rib isn’t supposed to be an even rib. Do k2, p2 on the 2nd row too and end with k1. You have 2 single ribs plus a garter rib.

Aa-Haa! I thought there must be some reason for this. Particularly as they are two different patterns and they both say the same thing no matter how many times I look at them.
I have another pattern for a short double rib scarf however, and that is worked on an even number with knit coming under purl and vice versa, in other words the way I think of ribbing.
I presume the odd K stitches at the end of every other row are to help when you sew the two sides together - right?
So are there two ways of doing double rib? Is one better than the other?

There could be more than 2 ways to do double rib. I’ve done a pattern calledl Twin Rib (from knittingonthenet.com) that’s a 3 st repeat, something like the scrunchable scarf.