Why you ignore the very next two sentences? De jure Crimean Khanate was protectorate of Ottomans. De facto they pursued their own foreign policy and treated as ally.
Regarding paying (ever increasing) tribute to Crimea the author references following source:
"Kaznu v Krym prodolzhat' prisylat' s pribyl'iu", Rossiiskaia Akademiia Nauk. Arkhiv, F. 1714, op.1, Novosel'skii, A. A., no. 66, 1.144
Regarding Astrakhan expedition:
"On their way home up to 70% of the remained soldiers and workers froze to death in the steppes or became victims of Circassian attacks. The Ottoman fleet was destroyed by a storm.
The Ottoman empire, though militarily defeated, insisted on safe passage for Muslim pilgrims and traders from Central Asia as well as the destruction of the Russian's fort on the Terek river."
(en.wikipedia.org/wiki/Russo-Turkish_War_(1568–70)
Looks like attrition claimed most of the casualties. The fact that Muscovy gave concessions despite winning a battle reinforces that.
Your tone sounds very neutral and well researched, indeed.
They were allies, true. But that doesn't imply that Muscovy wasn't paying tribute. Also once GH perished they broke that alliance.
"The treaty was concluded on 3 July (O.S.) / 13 July 1700[3] in Constantinople.[4] The Tsardom of Russia and the Ottoman Empire agreed on a truce set to expire in thirty years.[4] The sultan recognized Russia's possession of the Azov area,[2] including Asov and the newly built fortresses of Taganrog,[3] Pavlovsk[citation needed], and Mius[6] Russia dropped her claims to the Kerch Strait,[2] but was relieved from paying the annual tribute to the Crimean Khanate[7] paid since the occupation of Muscovy by the Golden Horde.[3]" (
http://en.wikipedia.org/wiki/Treaty_of_Constantinople_(1700))