So the issue was with dmxValidator. I've removed the validation rules for the form [it's a login form] but the validation was still there. Then I removed dmxValidator.js from site folder and reloaded it anew - fixed the problem so far.
Update! So the problem still persists. For some reasons, I can't upload it to remote server yet for you to check but I suspect the problem caused by HTML5 Form Validator. Even after I removed validation rules from AppConnect, the validations still run for the form. There are 2 things I tried to be able to get my login form to work;
1) By removing dmxValidator.js from the page at all 2) By enabling HTML5 Form Validator for my page from Insert > DMXZone > HTML5 Form Validator > Form Settings > Validate Form
I suspect there's conflict between HTML5 Form Validator and AppConnect Form Validator. I don't expect I still have to use HTML5 Form Validator since I upgraded to AppConnect Form Validator but validation for data exists in the database still requires HTML5 Form Validator installed and running.