Unlike most other digital cameras, Panasonic micro four-thirds cameras record lens correction information (distortion and CA, iirc) into the EXIF information of their RAW files (and will bake in the correction if you shoot JPEG). Some applications recognize and apply this correction information, some don't. That's why you're seeing different results, depending on which application you use. This is distinct and separate from the typical "lens profile" schemes that some programs use.

It appears that my original image has a significant amount of barrel distortion, why is this lens categorized as rectilinear in PTGui?

Because PTGui has a list of lenses that it can identify as either a rectilinear or fisheye lens, and barrel distortion alone isn't enough to make a lens a fisheye. Rectilinear basically refers to the projection of the lens. Fisheye lenses are designed to give an equisolid (or in rare cases, stereographic) mapping, rather than a rectilinear one.

The 14-42 lens is recognized as rectilinear, because it is rectilinear. And the amount of distortion changes as you change the focal length. And when the correction is applied, you may not see any barrel distortion at all.

More generally, why is there even a drop-down for me to choose between rectilinear and fisheye, given that it's purely a function of focal length, which is already an input?

Whether a lens is rectilinear or fisheye is not purely a function of focal length, but of design. PTGui offers both types in a pulldown because most folks use PTGui to stitch 360x180 spherical panoramas, and fisheye lenses are often used for their much greater scene coverage than rectilinear lenses, so you can get away with shooting fewer images and spend less time stitching. The field of view of the lens, which is important when trying to stitch images, changes drastically between rectilinear and fisheye designs. A rectilinear ultrawide can typically still require 20-30 images to cover a sphere, while a fisheye can cover it in 4-8 images.