Tire size compatibility, 700x35c or 700x38c for original size 700x37c
Both of those tires are quite close to the tires that you already have. There is a really good chance that they will fit. Because of the extra material in the SmartGuard layer these tires are taller than they are wide, but only by a couple of millimeters. So all you need to be concerned about is to look at the clearance on your bike with your current tires. Do you have decent clearance between the red of the tire and the crown of your fork and the bridges in the rear (or the fenders if you have the fenders installed)? 10-15 mm would be very comfortable, on my old Trek I've got like 5 mm. I don't have enough room to comfortably run fenders with my Marathons, but I'll be OK until the rain returns :-)

If you have enough room at the top, you probably also have room at the sides, but it is worth looking there too. The more generous the clearance the fewer hassles you'll run into with bits of junk on the road (it can get caught between the tire and the frame) or an out of true wheel – like if you break a spoke. From the look of the bike in the photos you should have plenty of room – the lower edges of the fenders appear to be aligned with the full diameter of the tires and the specs say that the bike comes with 37 mm tires.