"Resolution" actually means both things and in fact means also a few others, such as how many lines a lens can resolve at a certain focal length, etc.. That tends to be a big problem if you don't specify what you're resolving.
Most people in this forum, including myself, will use resolution to talk about the pixel dimensions of an image or screen. So Trevor is totally correct.
The DPI setting in the camera is only of benefit when printing the image or rendering it with a print intent. Neither number in the DPI of the photos is bogus however. I can't be certain why Fuji picked 180 specifically, but I do know why the first camera is set to 72.
"72" is the traditional screen pixel density and also how many points fit in an inch. It's typical to have screen-based images set to 72 DPI if for not other reason that to not have undefined or unusual (non-historical, non-common) numbers in the DPI field of the metadata. Most modern screens are actually much higher density than 72 DPI, or more correctly, PPI - since the "D" in DPI means dots, in colour printing it can refer to the individual droplets of one single primary colour, or is sometimes used to refer to a single primary colour in an RGB triplet. That's how you get huge DPI numbers on cheap printers and huge numbers on camera LCD specs.
The 180 value is likely the attempt of Fuji to set the value at something more meaningful for printing, which would leave you with a ~20x15" print, reasonable for that sensor. That's what you'll get without adjusting the DPI/PPI setting yourself either manually or through some other scaling operation, when printing/rendering with software that's aware of those settings, such as Photoshop or even Preview in Mac OS.
You can change that setting without altering the pixel dimensions at all to manipulate the printed size of the image. Some software will let you adjust the printed size directly in inches without touching the image data, so it's also really affecting only the DPI/PPI.
If you're going to print 4x6, reduce the number of pixels in your output images before you send them off to save time/space in file transfer. And possibly also be left with better output due to smoothing and other settings you can apply in software like Photoshop when scaling.