What was the first ocean the caspian sea lost connection with, beginning the process of turning what was once a gulf into an isolated body of water?

The correct answer is - Tethys Ocean.

The Tethys Ocean was an ocean that was situated between the European and African continents, through what is now the Middle East and parts of Central Asia, and the northern half of the Indian Ocean. Because of the geologic activities, mainly the movement of the Indian plate towards Asia, and the Alpine orogenesis, the ocean was becoming smaller and smaller, and it started to be divided into smaller pieces, until it totally disappeared. One of the remains of this ocean is the largest lake in the world, the Caspian Lake, often referred to as Caspian Sea, and it can easily be seen that this body of water was once part of a much bigger body of water with saline water, because it itself has saline waters.