One observation which may be relevant to some of this discussion. Sharpening, and especially evaluating sharpness, got more difficult for Mac users when Apple put 'retina' screens on their laptops (and now I think one of their iMacs). The high-res screens seem to me to make almost all photos look more punchy - sharper and higher in contrast at small scales - than they would on screens of lower pixel density. Presumably the intention is to make cell-phone photos look as good on your Mac as they do on the bright, sharp and contrasty screen of your cell phone. However I find that I cannot reliably evaluate sharpness on such a screen. Photos that looked great on my retina laptop screen often look noticeably unsharp on conventional screens, and remain off-sharp even when downsized. I would also add that trying to get around the problem by "pixel doubling" - enlarging to 200% so that each pixel in the photo covers 4 pixels on the retina screen - does not work for me. I don't find the 200% retina screen image equivalent to viewing at 100% on a conventional screen. When I updated my laptop about a year ago this forced me to do all photo sorting, evaluation and processing on an external monitor. For some this may be affecting assessment of sharpness in the full-sized image and in the down-sizing procedure.
Andrea, your 3 initial photos (in reply #8) look fine to me. The 3 downsized by Photo Mechanic look to me unpleasantly over-sharpened - i.e. distracting, over-contrasty detail at small scales, which prevents me seeing the photo as a whole. My eyes dance around the photos trying to see what's behind the distracting detail.
Werner (Alaun), your 3 look more natural. Perhaps they could even take a little more unsharp masking (especially the middle one, 1200_810_8171), but it's a matter of taste. With your procedure nothing about them looks unnatural or distracting on my screen.
I agree with John and previous posters, that if you know the size at which your output will be displayed, downsize to those exact pixel dimensions. Otherwise your photo will be re-rasterized to those dimensions by someone else's rendering algorithm, over which you have no control.
Cheers, John