[toni-a]

First printed sample bad... Ok for the paper and detail, but colors are definitely off: too warm. And of course I used the provided color profile.
Monday a new sample for another print service...

Is your screen calibrated ? most often with time screen colors tend to become  on the cooler side if no recalibration is  done, to see color accurate you push warmth, they print what they had.
Second eventuality is wrong color profile used : normally I send photos for printing in RGB or even sRGB profile to avoid them doing unexpected things for color profile  mismaatch, normally they set photoshop or whatever printing software on sRGB or Adobe RGB not a custom profile, and when they have color profile   mismatch warning they can either convert to working color space or use provided color space instead and here you might have issues . just send in sRGB or RGB and see f things go well