When you input a command in the shell, the shell is responsible for taking the whole string you typed:

curl -X POST http://localhost:4004/api/converter -H "Content-Type: application/json" -d '{"html": "a", "organization_id": "9213898231sadasd000"}'

and converting it to a command (curl) along with a series of options, separated by spaces. Most of the quote-marks used above are to help the shell split the values correctly. (see below)

System.cmd/2 operates at a level of abstraction lower than that - it expects the caller to provide the command as a string, and the arguments as a list of strings. Because of that, most of the quotes are not only not required, but could cause bugs. For instance:

-H "Content-Type: application/json"
# parsed by the shell as two arguments:
# -H
# Content-Type: application/json

But passing ["-H", "\"Content-Type: application/json\""] to System.cmd will send the double-quotes along to the program, which isn’t the intent.

For the input given, all of these quote-marks are formatting for the shell that isn’t wanted in the actual data sent to curl:

curl -X POST http://localhost:4004/api/converter -H "Content-Type: application/json" -d '{"html": "a", "organization_id": "9213898231sadasd000"}'
                                                    ^                              ^    ^                                                       ^

As noted in this thread from last week:

~w() will not do what you want if arguments have spaces in them:

~w(-H "Content-Type: application/json")

is a list with three elements:

["-H", "\"Content-Type:", "application/json\""]

You do not need the quotes inside quotes. These are just for shell, not for cURL