How to achieve 2 chosen hue colour separation profile?

One bad news. A lot of those images you are linking are not duotones... they are tritones. One Blue, yellow and black.

But let's see the color separations.

Using only one grayscale image is not good enough.

Take a look at this two posts:

Preparing design for duotone printing?

Printing photographs when job is a 2 spot color job

The basic idea is: take one "pure" channel, any of the RGB ones and use it as an ink. Use the opposite color.

On those two links I posted, I use one grayscale image as a base and one pure channel to do the "coloring".

But as you do not have a yellow channel I combined the Red and Green channel with multiply at the same time and used it as Blue. Then I used the blue as the yellow.

---

You also can tweak your CMYK color profile to reduce the amount of black used on the conversion.

Go to Edit>Color adjustments (May+Ctrl+K) And in the CMYK profile choose the custom profile. PLay with the settings there.

Then you can replace one ink with another color using multichannels.

You can just convert your image to a duotone in Photoshop.

See Use duotones in Photoshop - Adobe Support...

1. Convert your image to Grayscale (Image → Mode → Grayscale). You can use a Black & White adjustment layer first to control the conversion to grayscale if needs be.

2. Convert your image to Duotone (Image → Mode → Duotone). Select the correct type for the number of colors you want to use, set those colors and adjust the curves for each color until your happy with the result. Duotones are single channel images so if you need to create the separations yourself...

3. Convert your image to Multichannel mode (Image → Mode → Multichannel). This will separate your image in to the correct channels for each ink/plate/color.

A quick example...

Image 1: Original image

Image 2: Converting to Duotone

Image 3: Multichannel separations

---

^{Image source: https://www.pexels.com/photo/beautiful-blur-camera-capture-348528/}

In photoshop: Image > mode > multichannel

Then, in your channels panel you can add, remove and adjust channels. As for the exact result, you have to experiment some based on your photograph and the desired result. Also, read through the documentation on the website you provided - even without buying the color library profiles, the principles are the same.

Edit: A small tutorial

Set your image mode to multichannel. Go to the channels panel, where you now have Cyan, Magenta and Yellow. Double click each one to change the color. Try picking spot colors from the color libraries instead of using the picker. Experiment with the number of channels and their solidity. I came a long way just by changing the magenta channel to grey and making cyan a little darker blue. And remember, what may work in one photo, might not work in every one. I haven't worked much in this mode, but my guess is that every photo needs a bit of work.