A common way to make this look is to start with a black-and-white conversion, then add the blue tone as a separate adjustment rather than just cooling the white balance.

Two typical methods:

1. Add a solid blue layer above the photo and reduce its opacity.
2. Use a gradient map: map the darker tones to a darker blue and the lighter tones to a pale blue or blue-white, then try blend modes like Overlay or Soft Light.

Why your earlier edits did not match: changing temperature and blue saturation affects the original color image globally, while this style is usually a monochrome image that is then colorized. That gives a cleaner, more graphic blue tint.

So the basic recipe is: convert to B&W, apply a blue color layer or blue gradient map, then fine-tune opacity and blend mode until it matches the reference.