Cult, Sect and Denomination all mean the same thing.
To say any of the words is associated with *insert bad thing here* only means the media hasnt got a clue what it talks about. For instance, Christianity IS a Cult/Sect/Denomination of the Jewish Faith. And there are some who argue that Islam IS a Cult/Sect/Denomination of Christianity - though I personally would argue about the meaning implied by the word, after all unlike Christianity which grew out of Judaism, Islam did not grow out of Christianity - definitely dont remember any religious textbooks mentioning Mohammed being a Christian Teacher 
If a person is not familiar enough with the subject of the three words, which a lot wont be, why not just use Denomination as an overarching name?
However I would suggest that instead of immediately starting a subgroup and call it * Denomination* (For example) that instead when you create it you call it a Cult as it will simple begin as leaf on a tree, then as it gets larger enough to be a branch call it a Sect, and then once its ready to be its own tree, call it a Denomination.
You could also add in some underlying game mechanics:
A Cult could have at max 1 temple up to a Primitive Temple level. When it gains, say 10 nobles who follow it, it becomes a Sect and then it can have 5 temples up to a Medium Temple level. At 25 Nobles it becomes a Denomination and has access to Large Temples - say 15 temples. At 50 nobles it becomes its own Religion outright just because it has too many Noble Followers ergo too much influence. Would make things spice up as the main religion might try and suppress the Cult from gaining too much power etc. Would make for interesting intrigue and politics in religions.
The three words mean SIMILAR things. However there are subtle difference in their meaning and definitions. Couple of minutes reading the dictionary definitions would reveal that. The Jewish, Christian and Islamic faiths are all Abrahamic faiths, so they all share a common basis. The various groups that would one day give birth to what we recognise as the Christian faith were indeed once operated within the Jewish faith. It can't be a sect any longer, as it is separate from the recognised Jewish structures, which is what generally defines the difference between a sect and a denomination. There is no clear separation that would define when a denomination has ceased being such and become its own faith. One might argue that since Christianity and the Jewish faith disagree on several tenant that form the core of their beliefs, that they are no longer close enough in faith in practise to have one associated as a denomination of the other.
