More thoughts ..... on the first half of the pattern (up to row 12) all the K8, P8 rows are worked on EVEN rows.
The switch comes, as you noticed, on Row 13. From then on, all the K8, P8 rows are worked on ODD numbered rows.
I don't think or at least, I can't see, where you're going wrong at all.
If you break the pattern up into bits, this is what you'll find:(Bottom Right of chart)
Rows 1 to 12 - there are 12 rows of solid
white - Knit on Right Side - Purl on Wrong Side.
(Top Left of Chart) There are only 9 rows of solid
grey - Purl on Right Side - Knit on Wrong Side.
Although there are 20 rows of the pattern - it is not equal because there are more Knit on Right Side than Purl on Right Side rows.
I have a similar pattern in the photos below- I've knitted the back and one front of a cardigan and it does not lie flat. If you look closely at the orange wrap in your photo, the stitches are not lying flat either. I don't think it's a pattern that's meant to look even (that's my story and I'm sticking to it!
You'll see in the second photo I've posted some very uneven stitching on the left of an oblong (second column) which I frogged. It's important to keep the tension very even on a pattern like this because of the total difference in the stitches - the change can soon look floppy if there's a bit too much slack.
Anyway, thanks very much for posting, Cooks - it's given me the kick I needed to carry on with cardi.
Hope this helps.