Hi, confused user here. 
The OP is making a PUT request, and their fix was simple enough (even if @crashandburnuk very correctly commented that this is not really very intuitive).
I have also found out that things are a bit easier when making a POST request. In that case, Postman does respect the Content-Type header. At least, that’s what I’ve found out. That’s true even if the payload on the body is empty (length zero), which technically would mean that the Content-Type would not apply (on the other hand, one might argue ad infinitum that a payload of zero bytes can have any Content-Type — since zero bytes is a valid length for a payload 
)
Now… how to accomplish the same with a GET request…?
For GET, Postman allows a payload to be sent (which is not forbidden) and so it follows that it should be possible to set the Content-Type as well. However, Postman (as well as others, such as — allegedly — .NET) does not allow that type of configuration.
curl has no problem whatsoever, it’s totally agnostic to what you do or don’t do, so long as it’s not explicitly forbidden by the specs.
According to the specs, if one wants that the reply to a GET request to explicitly be in, say, JSON or XML (as opposed to the default, whatever that might be), one ought to place Accept headers indicating the content-type preference. Sadly, though, many (most?) APIs simply couldn’t care less about Accept headers, and just send whatever the server wants — unless the request does, indeed, specify Content-Text… which is not possible to configure in Postman for GET requests.
It seems to me that there is a slight inconsistency on Postman’s interpretation of the specs. Either GET cannot have a payload, and then the request body should be greyed out (and no Content-Type header is sent); or, like curl does, if a payload is allowed with GET, then setting the Content-Type should be possible, too — even if the body is empty!
I’m not an expert in interpreting the specs, mind you. Exchanging a few cool posts with Tim Berners-Lee when he announced his ‘cool new idea’ in the early 1990s does not make me an ‘authority’
(not really worth taking a look, but here’s the link to my public workspace)
