I count 13 elements on the Battelle antenna. However, the driven element looks very much the same. Note the round thing on the folded dipole at the top of: and compare with the eBay photo. Also, note that the mount is very much the same. However, there are no vinyl caps covering the elements, so they're not exactly identical. Close enough, methinks.
This looks like the "GPS" antenna in its radome:
Maybe, but there's another problem which I didn't mention. The elements are designed for aesthetics, not function. In order for each pair of elements to be the correct 0.5 wavelength, the length would need to include the boom diameter. On the one I bought and returned, the length did NOT include the RF path around the outside of the boom resulting in gross mistuning. I could buy one of the pictured eBay antennas and make some measurements, but I don't need another misdesigned antenna.
Incidentally, in order to match a 200-300 ohm folded dipole, a 4:1 coaxial balun is required. Something like this: There's not enough room inside the round thing on the folded dipole for a balun.
Even reputable companies make mistakes designing Wi-Fi yagi antennas: MFJ didn't have the boom path problem, but did manage to design a 200 ohm antenna instead of 50 ohms. They then used a ferrite bead to "fix" the VSWR problem.
GPS is RHCP (right hand circular polarization). It would work with a linear polarization, but would be orientation sensitive and have a
-3dB polarization loss.
Same here. Hard to tell without numbers but methinks my guess(tm) is fairly close to the mark.