I honestly don't have a lot of experience, but I've studied the terminology and some of the psychological background.
Those truly transgendered persons want to be called by their true gender ID. MtF = Female, FtM = Male and there is also the gender-queer status these days or may referred to as gender-fluid. (Basically, not a binary gender to one or the other.)
Those terms such as New Half, Ladyboy, Shemale -- you'll see more commonplace in sex workers, porn or something with a sexual fetish. However, in Japanese society... New Half is basically an accepted term to describe a transgendered person. Still, in reality, that 'New-Half' still wants to be just described as Male or Female.
CIS-Gender (biologically born male or female by birth) people fall into three categories:
1) Don't understand, Don't care and "Burn at the stake" mentality.
2) Accepting but offensive by not caring to understand the terminology.
3) Educated, accepting and has no boundaries to interact with all sexes of people.
Surprisingly... cis-females seem to be on the extreme on both ends of the scales. While cis-males, while solo can be quite open-minded and accepting... take that same guy and put him into a group, that mindset changes and not always for the better.
Oh, lastly, whatever the terminology... just because a guy becomes a girl or vice versa, that may or mat not change their sexual orientation. A huge misconception especially among MtF persons.
Yeah, probably wrote too much... but take time to educate yourself if you interact with anyone who fits into any category.