Natural dark hair. How can I go from a warm to a cool tone?

I have natural dark brown, almost black hair and the last time that I have bleached or dyed it was many many years ago. As most brunettes, my hair shows a reddish undertone. And I want to get rid of it. Plus, I have understood that using a semi once in a while can even have a positive effect on the health of your hair.
I love how grey hair looks. I would like to cancel the warmth out, make it cooler and get close to that grey hair.

I went to the hairdresser where they told me that in order to get that greyish tone, they had to bleach my hair. As I do not want to change my hair colour, but make it cooler, I rejected the offer. Then they tried to offer me dying my hair in a brownish colour. Again, I rejected the offer because I know that a brown hair dye will not help me to get rid of it but make it worse. 

I have looked up some possible solutions and I would really love to get your advice:

1. In terms of nourishment, pigmentation and duration, should I use Makki colouring mask (Grey) or Nirvel Artx (Grey)? If your have better suggestions, please tell me!

2. Also, Makki recommend on their site to add a bit of blue to get a cooler look. Should I do this? Or should I use green? Or a bit of both? I have no experience so a bit unsure here. I have read into how hair works. If I understood correctly, green would cancel red whilst blue would cancel orange. I am not sure if my case I should see my hair undertone as orange or red. What would you say? I have uploaded two pics of my hair, one in natural light and the other in artificial lighting.

3. I have seen and read about diluting the dye with conditioner. Should I do this with the grey?

Thank you!

xxx from Amsterdam

- Pic 3 (girl with red lips) is an example of what I want to achieve.