Terri, I believe you'll find that you've got these terms exactly backward. If you google "petersham ribbon", you'll find dozens of sources, articles and photos confirming that the material at the top in your photo is typically called, and is commonly sold as, petersham, or milliner's petersham, (as labeled). Also, it's uniformly maintained that it's petersham that can be shaped, not grosgrain.
Eg:"This is a stiff ribbed ribbon, used to reinforce waistbands, bind corset edges or hat brims, or even as a foundation for neck ruff. ... Petersham is often confused with grosgrain ribbon but the important difference is that the petersham has a scalloped edge which allows you to shape the ribbon. Usually made out of 100% Rayon."
It's my understanding that "grosgrain", properly used, is a
technical term describing the heavily-ribbed (gross-grained) appearance of fabrics (not just narrow ribbons) that comes from the different weights of the yarns used (heavy weft, light warp) in the weaving. Thus the term encompasses what is commonly called petersham, but in current regular use, esp. in the ribbon-craft world, grosgrain is more widely and loosely used to mean any ribbed-woven ribbon, as opposed to satin-woven, jacquard-woven, or non-woven types, and includes ribbons with stitched, reinforced or complex selvedged edges that preclude any possibility of shaping the ribbon. "Petersham" is used in various trades to specify gross-grained woven ribbon with a simple unstabilized selvedge that does allow the material to be shaped (it's the heavy uncovered weft that gives the "scalloped" look).
The material at the bottom in your photo is thus a complex type of grosgrain (since it's got a crosswise rib), but isn't petersham because it's got stabilized edges. I'd call it a type of "waistband stabilizer", since it appears to be specifically made for that purpose. I can imagine it being referred to as petersham in some shops or workrooms, and some general dictionaries simply define petersham as "a corded material for hatbands, the insides of belts, etc., or a narrow belting for the tops of skirts." But that's not really a precise enough definition for our needs…
However ill-informed my understanding of the history and manufacture of these items may be (I'm certainly no expert!), in practice and when searching out the stuff at the top in your photo, or any shapable ribbon for tailoring, millinery and other garment-related projects, I believe one will do best to use the term petersham. If they're looking for the stuff at the bottom, I'd say ask for neither petersham nor grosgrain; ask to see the waistband stabilizers.
dpc