Adjust the Exposure setting as required so that the central area of both images is identical. Open both images in LR and adjust the raw image using the Basic controls to match the JPEG image in tonality as close as possible. Make sure the camera is perpendicular to the subject so that there is no convergence (horizontal and vertical lines parallel to the viewfinder frame) Use maximum aperture and widest zoom setting, which is probably worst case for both vignetting and distortion. Shoot a large "evenly lit" 2D subject with linear detail, such as a brickwall. It should be easy to check if LR is applying any lens corrections by comparing the JPEG and raw images inside LR: Both of which can be manually enabled or disabled.
The only way I'm aware of that LR can apply lens corrections is with the Profile and Color panels in the Develop module. What a difference one-letter makes! At least we know the available Adobe lens profile works fine with the predecessor X100 raw files. With a better example X100S raw, maybe shot at F/8 as the X100 sample is, it would be easier to see a difference between the raw-digger and LR conversions. My guess would be that Adobe does what the camera does, and if there is no way to turn off the lens corrections in the camera then Adobe has them automatically enabled all the time, like they do with many other point-and-shoot and mirrorless cameras, nowadays. My analysis, above, comparing the RawDigger (dcraw) raw conversion which shows a slight darkening in the corners and the LR conversion which does not, is that vignetting is being corrected by LR automatically, but there is so little geometric distortion in the test raw I found that I can’t tell one way or the other about that correction.
One such piece of software is PTLens, which includes the Samyang 14mm f/2.8 as one of it's profiled lenses.These statements make sense about the X100 which has a lens profile in LR that can be turned on and off, but we’re talking about the X100S, not the X100, and there is no X100S lens profile to turn on and off so one cannot tell whether LR is always or never correcting the distortion by using LR, alone. In this case you can rely on someone somewhere figuring out the set of parameters (usually called a profile) and posting them online. If your lens is fixed focal length and the distortion is independent of focus distance (which I think it largely is with the Samyang 14mm f/2.8) then one set of parameters can easily be applied to all images generated by all copies of the lens (that aren't decentered in some way). Therefore you need a piece of software/plugin that can do different types of distortion, along with the parameters of the distortion. You can usually only tune one parameter of the correction so you'll never eliminate the moustache.
The reason the Photozone review states that this distortion is difficult to detect, is that most software only offers very simple correction based on radially symmetric distortions based on simple formulas. It's very difficult to correct by hand, but very very easy for a computer to correct, given a formula for how the distortion behaves.