If gender is to be useful as a tool in communicating it needs to be based on commonly understood concepts.
Quite frankly the fact that trans women want to be referred to as women is a perfectly understood concept.
In my example with color, if we used mutually exclusive words to describe the same color the the concept of color will become meaningless.
This is a false analogy. I could, for example, provide an alternate analogy where it's you who insists on calling a pen that's been painted black red because it's the original color.
If you separate those then the words "man" and "woman" will stop having meaning
No. The word "man" would refer to people who identify as a man and who are at some process in transitioning. If you're polite it could refer to people in their pre-transition stage too. The word "women" would be similarly defined. We already have biological sex specific words: "male" and "female".
we need to have those concepts for effective communication.
Why? When I talk to somebody on the street/ in the workplace, I don't care what their chromosomes look like. If they look like a dude I'll call them "man". If they say "No, I'm a girl" I ain't gonna perform extensive DNA analysis. I'm going to take their word for it. I don't see why their chromosomes matter. (and what about people with XXY chromosomes, or people with XY chromosomes but with the Y chromosome deactivated?)
And the idea you brought up that someone will suicide themselves if they are called a standard word, but one other than they would prefer is ridiculous.
While you may see someone's biological sex as a more important distinction to make than their gender, I don't see how you could honestly fail to understand why calling a MtF trans person a man is hurtful to that person. Imagine, if you will, a feminine man. At first glance he may look like a woman, so you call him a woman. However, he says he's actually a dude. If you, after being pointed out to their dudeness, refer to him as a 'her', he will get pissed at you.
If someone is suffering because of this they have other, deeper problems
Yes it's called gender dysphoria and the recommended treatment is transitioning. The recommended way to support them is calling them the gender they've transitioned to/ are transitioning too.
Even if separating gender and biological sex is abhorrent to you, I fail to see why just placing more value on a person's gender when deciding which pronouns to use is such a problem.