My guess is the 37x622 is the precise ISO/ETRTO size while the 700x35C is the approximate French designation. See:
https://www.schwalbetires.com/tech_info/size_markingsAt this link, Schwalbe states...
The ETRTO size specification 37-622 indicates the width of 37 mm and the tire inner diameter of 622 mm. This dimension is clear and allows a precise classification of the rim size.
A bit later in the page, they say...
French size markings (e.g. 700 x 35C) give the approximate tire outer diameter (700 mm) and width (35 mm). The letter at the end indicates the inner diameter of the tire. In this case, C stands for 622 mm.
It is useful to read the entire article at this link, but this explains the seeming mismatch in your Schwalbe box marking. Makers sometimes differ in their sizing conventions, as this useful article by Sheldon Brown states:
https://www.sheldonbrown.com/tire-sizing.htmlJust out of curiosity, what is their actual, inflated width of your tires when installed on your specific width rims, Mike? A rim that is wider or narrower than optimal can affect the width of a mounted tire to a degree.
I've also had tires "grow" a bit over time as the casing stretches, sometimes by as much as 2-3mm in actual width in the course of a week or so. For example, a pair of Panaracer Paselas I bought were labeled 700x37C, measured 35mm wide when first mounted, and "grew" to the stated 37mm after a week inflated on the rims. EDIT: I just checked and in my case the ISO/ETRTO and French markings on these Panaracer tires match: 37-622 molded into the tire and the sidewall label says 700x37C. I no longer have the Panaracer box to check against the tires, as you do with your Schwalbes.
Best,
Dan.