Comparisons, always a bit dubious, you show your comparison and I will show mine

Worth trying other applications, especially Krita and Imagemagick.
Attachment:
cmyk-test.jpg [ 174.92 KiB | Viewed 3155 times ]
You did not say, (edit: You did say, but worth checking anyway) but are
all the parameters the same and what are they? Worth checking using ImageMagick
identify -verboseThe above, conversions all from sRGB to ISO coated V2 and perceptual, although PS is little dubious with ISO coated (PS = Euroscale V2???)
Looks ok to me but at the moment using a linux laptop (Kubuntu 16.04) with very little colour management.
For info, the web conversion rgb2cmyk.com uses GraphicksMagick.
Subjectively, I think the ImageMagick conversion is closest to the original RGB.
convert RGBtest.png -profile srgb.icc -colorspace cmyk -profile ISOcoated_v2_eci.icc IM_cmyk.tif
In the end, largely irrelevant, CMYK is for the printing company and they will screw it up anyway.
edit: Looking at your last post, very artistic, Krita might be a good choice since you can start a project in CMYK mode. No need for a conversion.