You didn't use the metric, the map is accurate.
And it's such a lovely metric that certainly does work on that map.
Trouble is, it's all well and good to bandy about words like "non-euclidian" because any observer in such a space would perceive it as normal, right?
This makes it a terribly nice idea, but a nicely terrible fact - sure it can be explained, but it can't be proven.
I think if you proposed this to any geophysicist or mathematician, they would be much happier with the statement that "a metre in Toronto is the same distance as a metre in Johannesburg and a metre in Sydney" and so on.
And once again, the Flat Earth theorists can't say math is the instrument of the devil when it proves them wrong and it is the word of Holy messengers when it helps them explain their own models.
Religiously hyperbolic of course, but you get my meaning.
You can't have it both ways.