Governments are different as they legally have to do it.

I don't think it's illegal requirement, they just choose to respect the wishes of the receiving government.

Of course. The will (and should) use the name the country officially defines to be the correct one. Why would they refuse and cause a diplomatic fuss over such a trivial thing?

This is AFAIK a result of trying to get Erdogan to accept the NATO application from Sweden and Finland.

NATO head Jens Stoltenberg started saying Turkey but then changed to Türkiye before the big NATO summit in Madrid last summer.

Interesting. From the State Department:

"Note: The official conventional long-form and short-form names remain “Republic of Turkey” and “Turkey”, respectively. “Republic of Türkiye” should be used in formal and diplomatic contexts. The conventional names may be used in place of or alongside “Türkiye” in appropriate instances, including U.S. government cartographic products, as it is more widely understood by the American public."

Source: https://www.state.gov/u-s-relations-with-turkey/