Ie they probably lied. The fact they kept an imperial title is telling.0
You still did not quite get it and I admit that the issue is somewhat convoluted, especially for an outsider so let me try to add some "depth".
The 1st "imperial" claim dates back to the reign of Ivan III who did not even officially adopt tsar's title but was married to a Paleolog and combined this linkage to the Byzantine empire with the older claim going back to Vladimir Monomah. Ivan IV was the 1st Russian ruler who became Tsar officially and he and his successors had been crowned by "Monomah's Hat", the crown of Byzantine origin. The
diplomatic argument was that Tsar is the same as "Caesar" and as such is equal to an emperor. Needless to say that this claim never was accepted by the Hapsburgs even if from time to time some of the Western diplomats had been using "Imperial Majesty" (IIRC, this was, for example, the case with the British Ambassador at Peter's court who in his speech made in 1710 used the imperial title; at least this was claimed by vice-chancellor Shafirov who was at that time in charge of the Russian foreign policy).
After the Peace of Nystad Russian Senate and Synod came with initiative of the brand new title which was intended to identify new Russian position in Europe. Peter was declared an Emperor of All Russia. Actually, initially Peter was somewhat reluctant to accept because of the existing legacy. However, eventually he was persuaded with an argument that "Emperor" is a
Western title which reflects the new position of Russia as the victorious
Western state too big and powerful to be just a "kingdom". As a part of the argument, it was assumed and agreed upon that, contrary to the earlier claims, "Tsar" is just an equivalent of a "king". This had some supporting logic because the full title at that time included ".... Tsar of Kazan, Tsar of Astrakhan, ... Tsar of Siberia, ..." (none of them having any imperial connotation), and fit the notion of an emperor as "Overlord of the Kings".
In other words, the old meaning of the "Tsar" was officially downgraded to be replaced by the Western "emperor/empress". The same happened with the coat-of-arms (Monomah's Hat was replaced with the imperial crown), crown, scepter, etc. In other words,
this imperial title represented a complete break with the old tradition and its claims. It does not make sense to talk about "lying".
Now that I didn't know. Thank you for pointing out.