I'm a bit confused about your question. Natural light v Light emitting diode is not an option to choose between. leds come in a range of colours, halogen lights are much more (10x) expensive to run and I think there is less colour choice.
This article explains about light bulb colours
https://www.led-lighthouse.co.uk/warm-white-natural-or-pure-white
I've was interested in your question about colour blinded ess and I've read through
http://www.colourblindawareness.org including their advice to business and retailers. Nowhere does it suggest a particular light temperature to help colour blind people. I suggest that a natural or pure white light, rather than a yellow or blue light would be the most accessible, bearing in mind that there are several types of colourblindness and these affect different colours. It's not always red and green that are the problem shades. I think it would be better if you gave clear advice, explanations and reassurance about what tones and shades suit and what are harsh against skin tone. The website gives a lot of very helpful information about accessibility, you could always contact them for advice about hair colouring.
Moving on to the rest of your question about lighting, Let's break down lighting into 3 main aspects. You need to be able to see well, clients also need to feel good about themselves so the lighting colour needs to flatter skin tones. You may wish to your lighting to add atmosphere and mood.
To see well you need to create a good level of general illumination. Lighting is measured in brightness of the light bulb (lumen) and brightness on the surfaces (lux). You need a minimum of 500 lux in a retail space. 1 lumen is a measure of the light from one candle and lux is how bright 1 lumen makes 1 square metre from 1 metre away. If you measure the surface areas of the salon, you need 500 lumen for every square metre of surface. For big commercial spaces there's a more precise calculation, including the beam angle from the lightbulb and the cone of light that's emitted. But basically you want a grid of ceiling lights not more than 1 meter apart all providing a minimum of 500 lumen, or the equivalent.
Old fashioned light bulbs are measured in power consumption or watts. We know from experience how bright a 40W, 60 W and 100w bulb make a room but the important factor is the brightness or lumen not the power consumption. When choosing energy saving bulbs ignore the watts and look for the lumen.
To convert an old fashioned light bulb to an equally bright energy saving bulb you need to match the lumen.
Rough conversion for old fashioned incandescent light bulbs
40 Watts (W) = 450 lumen (lm)
60 W = 800+ lm
75W = 1,100 lm
100W = 1600 lm
Fitting a grid of recessed spotlights in your ceiling is expensive and lighting panels are not very flexible if you don't like the end result or want to have a bit of a redesign. So the most common retail installation is track spotlights on ceiling or suspended tracks. This type of light fitting will be cheap to install and run and very flexible if you rearrange your layout. By angling the lights carefully you will get very good quality overhead lighting without flicker, glare or shadow.
At this stage you don't need to specify the light type that you want, just that you need an overall illumination of 500lux. You can chose whatever lamp shades you want but just ask for GU10 fittings. You might save money by having a boring fluorescent strip light or ceiling panel light in work areas here and there so that you don't need to buy loads and loads of spotlights.
The GU10 specification is the connector that goes in the socket to take the light bulb, it has nothing to do with lighting colours or anything - it's the most widely available fitting and makes life easier when it comes to buying light bulbs. . .
Once you've chose your light fittings you then need to think about the light bulbs and what colour of light you want. You can get everything in led, you don't need to choose more expensive to run halogen.
Your clients will have warm light bulbs in their homes most probably and cool lightbulb at work. Cool lighting is less flattering but closer to actual daylight. The pros and cons of warm temperature and cool temperature lighting is discussed in the article link below. Whether you choose warm (yellow) light, "natural" or "pure" white light or cool "blue" light you need to choose light bulbs with a high colour resolution (CRI) above 86. So not the cheapest lightbulb in the wholesalers!
https://www.salontoday.com/375294/six-things-you-need-to-know-about-salon-lighting
Once you've got your light bulbs inside your lampshades you can play around with how close together you need them and what angles you like. You might want to shine some on wall displays and create areas of light and shade. Just make sure you have at least 500 lumens shining right over each cutting chair.
You'll also need wall lighting to avoid casting shadows on client faces - so you'll need some lighting each side of your mirrors. As warm light is more flattering you might want to choose this for making clients feel more comfortable. You might want to add some up lighters to create lighting effects on the ceilings and walls, or have some standing or table lights to create comfort in client waiting areas.
In our salon and we've gone for natural light with a temperature of 4000k. At first I included some warm light bulbs but I was advised to choose one or the other. I switched to natural except for wall washers illuminating a natural stone wall. This looks much nicer with warm lighting.
I'm still tweaking the lighting in my salon, I inherited spotlights on tracks but with halogen bulbs in. I swopped them all for leds which saved me loads of money and I read up on lumens and so on so the end result was much brighter and more appealing than the previous tenant. However we had a TV crew use our salon as a backdrop for some interviews a while ago and they used a big bright photographers lighting panel and wow, place looked amazing on TV. So upgrading my lighting with extra spotlights is on my To do list.
