I found that the problem is in the Basic Auth plugin. WP-API guys recommend using their own plugin and this solution works for me.
Deactivate all activated basic auth plugins in your WordPress dashboard
On the machine your WordPress is running go to the plugin folder
Run
git clone https://github.com/WP-API/Basic-Auth.git
Go to your WordPress admin dashboard, plugins page. JSON Basic Authentication should be in the list. Activate it.
Now creating a record via POST request should work.
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