The only "Carbon base" I can think of is things like fumarate, acetate, etc. But that is hardly carbon, there is an oxygen too as it is a carboxylic acid(I think)
Yeah you call that organic (well unless with exceptions like an inorganic carbonate salt / complex)...
Carbon base is rather a phrase like "carbon based lifeforms" (good album by the way), which just means that the key element in the organic tissues is carbon rather than say silicium, although there don't really seem to be other good candidates and even silicium has a lot of limitation compared to carbon. Life on earth is carbon based.
All of this is making it even more complicated though...
They are just called freebase or salt form, the types DMT and other alkaloids come in.
As said, freebase DMT is commonly found as it vaporizes (smokes) well.
There is nothing to 'prefer': depending on how you want to take your DMT or store your DMT one form or another is just objectively more ideal. It is not a subjective preference, and the effects are not different.
Freebase DMT is good for vaping / smoking, but not useful for other routes of administration and it is less stable.
DMT salts (some even more than others) are more stable. For injecting you need a salt, for plugging or snorting you want a salt which will burn less and absorb better. For oral use you want a salt like the phosphate, however ingesting the freebase will automatically make it turn into the hydrochloride salt in your stomach because of stomach acid.
(free)base + acid --> makes a salt
There are many different salts possible of DMT because there are so many different acids you could use to convert your freebase into salt.