The linked map is somewhat imperfectly trying to convey the complicated geopolitical situation that typified Persian politics in the decade prior to the First World War. The 1905 Constitutional Revolution in Tehran in which the Qajar Shah, Mozzafar-al-Din Shah, had to accept a constitutional government deeply alarmed both the British and the Russians, both of whom had interests in Iran. Mozzafar-al-Din Shah's successor plotted with both the British and the Russians to undo the constitutional limits on the monarchy's power. Meanwhile, the British and Russians decided to split Persia up into spheres of influence, and here is where the OP's linked map came in. The Russian half matches the convention's division of the country. Of course, carving up a nation into spheres of influence is more easily said than done and Persian-Russian relations between 1907-14 were highly confusing. Although the Shah was able to use British and Russian military aid to bombard the Iranian parliament, the Majlis, in 1908 and try and dissolve the constitution, the following year, constitutionalist forces were able to regroup and forced the Shah to abdicate. In 1911, the Russian government issued an ultimatum to the new Persian government recognizing Russia's special interests in the north as delimited by the 1907 Anglo-Russian agreement. Even tough popular opinion and the Majles supported rejecting this Russian ultimatum, the central government relented. Russian forces occupied the north of the country up through the outbreak of war. The Russians supported seemingly pro-Russian forces, such as the Persian Cossack brigade, which had led the 1908 attack on parliament, and staffed it with Russian officers. This brigade expanded to division-size during the war became one of the more powerful forces operating in Iran between 1917-21. The British replaced the Russians as the main patron of the Brigade after the Russian Revolution and promoted an officer named Reza Khan to command it. Reza Khan would use the power of the Brigade to national leadership and became the founder of the Pahlavi dynasty in a 1921 coup.

So while Persia was a nominally independent state on paper in 1914, its sovereignty had been deeply compromised by imperial interventions into Persian domestic politics.