Help! Fisherman’s run in the round to back and forth?

Hello! Im a bit of a beginner knitter (first time knitting in the round!) and need some help please. I’m making a sweet little jumper for my nephew but can’t seem to get the knitting to look the same from in the round to back and forth. In the round, I was knitting alternating (Row 1): knit, (Row 2): P1, knit into the stitch below. For back and forth, I did (RS): k1, k1below; (WS): k1below, p1. The stitches in the round feel ‘smooshier’ and tighter than back and forth. What should I be doing for back and forth rows so it all looks the same? See photo.
Thank you!

Here is a video of fisherman stitch (aka shaker stitch). He does it back & forth.

Now, if doing it in the round, you’d be doing it like (his) row 2 all the time? Right?

If anybody sees I’m wrong please step in and correct me. But it’s what I would do.

FluffyYarn, You’re not wrong but I think the difference here is that this is half fisherman rib rather than the full-on fisherman rib. Even at that there are some variations in the way it’s knit.

If the knitting in the round is:
(Row 1): knit,
(Row 2): P1, knit into the stitch below.

then knitting flat should be:
(Row 1, wrong side row): purl
(Row 2, right side row ): P1, knit into the stitch below.

Welcome to the forum,HannahMill!

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Thank you!

This also depends on which is you last RS row (1 or 2)?

If your last RS row in the round was K and your front has an even number of stitches then back and forth would as follow:
WS: (P1 below, K1) repeat.
RS: K across.

With an even stitch count:
WS: K1, (P1 below, K1)