I've always been told in school that these leaders used cults of personality to force ideology on their nation, but my own readings have brought me to believe that they were much more organic expressions of genuine support from sections of the people.
To what extent are each of these stories true?
I'm not sure how such a cult could ever be "organic expressions of genuine support" from below. Whatever about popular support for the Soviet regime (which remains a contentious topic), the organs through which the Stalin cult was propagated were always tightly controlled by the Party. That is, newspapers, party cells, rituals/parades, portraiture, theatre, congresses, literature, etc were never neutral conduits for popular sentiments to be aired in. Such media/channels existed to propagate the official discourse, not provide a platform for popular opinion. After all, this was a dictatorship.
Nor would the assumption of "genuine support" explain the evolution of the cult. Prior to 1933 it had tended to focused on the leadership as a whole, portraying Stalin as merely first-among-equals within this collective leadership. From 1933 onwards it quickly shifted so that Stalin's sun eclipsed all other Soviet leaders; it's only at this point that the more ridiculous sentiments, such as 'Stalin is love', begin to be aired and Stalin himself is imbued with superhuman powers. This was a response to the shifting political realities within the Communist Party, something that was not readily visible to ordinary workers or peasants.
What this doesn't mean, of course, is that the cult was planned and executed by a shadowy cabal of Soviet spin doctors at the very top of government. But those "sections of the people" who had a role in developing and encouraging the cult were the same writers, journalists, artists, etc, who relied on the state/Party for patronage and direction. Rather, it's worth speaking instead about the emergence of a Stalinist intelligentsia and how that shaped social discourse.
Which is not to forget the Party itself, naturally. A local Party report from 1936 gives a taste of the, slightly hysterical, role of that this played at all levels of society in pushing the cult's messages:
During agitation and propaganda in the press there must be more popularisation of the vozhdi [Party bosses]. Love for them must be fostered and inculcated in the masses and unlimited loyalty, especially by cultivating the utmost love for comrade Stalin and the other leaders amongst children and young people, inculcating Soviet patriotism and bringing them to fanaticism and love and defence of comrade Stalin and our socialist motherland.
This was after all one of the key purposes of the Party and its extensive apparatus: control. The messages that it, and the media, publicly distributed were dictated from above. This includes the personality cult.
Sources: See Sarah Davies' Popular Opinion in Stalin's Russia and Shelia Fitzpatrick's Everyday Stalinism.
What are your "readings" and sources supporting your belief that it was organic expressions?