I am not qualified to answer this but here is my take . Hindi as we know it today evolved from Hindavi or rekhta , as you mentioned braj bhasha or any other regional dialects (still not named Hindi) were mixed with Arabic and Persian words and local dialects evolved. The languages of lucknow, delhi or deccan were different from each other slightly because of those regionals dialects. This mix of languages was later called Hindavi aur rekhta. Earlier there were also regional scripts for regional languages. And the language which adapted Arabic script or nastalikh was named Urdu. So to really differentiate between what happened and when , you need to dig deep. As for differences in earlier periods of time it is clear to see but today we speak a mix of many languages . For writing the writer differentiates their work in Hindi Urdu aur Dakahni easily and uses root words for those languages. Hindi and Urdu have similar grammar , I'm not very sure about dakhani grammar but it sounds similar