The reason we disagree is because at some point, we ask, "Why should we treat you like a man?"
1. Because I want to be treated like a man, live like a man and do my best to be a man regardless of whether you believe me to be a real one or not, and there is no good reason not to treat me like a man. (that's good enough for me, actually)
2. Because I have a legitimate medical condition the accepted treatment for which is to treat me like a man regardless of whether you agree that I am a real man or not, and there is no good reason to treat me like a man (that really IMHO be enough for anyone)
Note that I am leaving open the 'there is good reason not to' window here - that then easily allows us to discard nonsense objections like 'then should you get checked for testicular cancer?' and to have discussions around less nonsense one's like 'should we allow transmen to step in the boxing ring with cismen'
When it comes to playing on a sports team, we have historically had a "women's" competition. If we take your first suggestion, and eliminate the two "man/woman" categories, and replace them with four "cis/trans//man/woman" categories, we have to change the way we do sports.
We don't necessarily need to change anything. We can just assign the 4 groups to the existing categories for sporting purposes. We can change things though. Sports already have their own rules and regulations and definitions. Each sport may even have a different definition. As the Caster Semanya case shows it's not necessarily a trans-issue alone. I think each sport is going to have to navigate that one differently, and possibly even different levels of sport will have different regulations.
For sure, it has been shown that it is 'possible' for transpeople to compete in sports as their preferred gender without causing significant problems. Equally I have sympathy with the idea that it is not always possible for this to be the case.
That's true, but I would be curious what you think the implications of the statement are. I think the implications are that the existence of the penis is actually more important than the words used to label the person attached to it.
You could be right. But that would reinforce my point that it is not about the words or definitions of male and female. We may be in violent agreement at this point.... the question is not whether people with male biology can be women but whether it is the right thing to do to treat those who identify as women as women.... and in what cases exceptions might have to be made.
And none of that question is addressed by chasing shadows of nitpicking definitions of man woman male female.