To pay for a spell with C in its mana cost, such as Kozilek, the Great Distortion, you have to pay colorless mana (C.R. 107.4c, 601.2f, 601.2h). You can add colorless mana by tapping Kozilek's Channeler for CC just as well as by tapping Ruins of Oran-Rief for C. (Compare C, which means one colorless mana no matter how it was produced [C.R. 107.4c], with S, which means one mana of any type produced by a snow source [C.R. 107.4h]. Also compare C with 1 and other generic mana symbols [C.R. 107.4b].)
Note that Eldrazi is a creature type (C.R. 205.3m) with no rules or game mechanics inherent to that type. Note also that not all Eldrazi spells have C in their mana costs (see, e.g., Eyeless Watcher and Kozilek, Butcher of Truth).
EDIT (Feb. 4, 2021): Edited to conform to rule update for Kaldheim.
So the 1 mana symbol counts as a diamond symbol? That’s what I’m getting
1 means one mana of any type (C.R. 107.4b), while C means one colorless mana (C.R. 107.4c). For example, 1 can be paid with W, U, C, or one mana of another type, whereas C can be paid only with C (one colorless mana).
Some cards formerly had the text "Add 1 to your mana pool". In general, those cards now say "Add C" instead (C.R. 108.1). Similarly, "Add 2 to your mana pool" is now "Add CC". However, this doesn't affect 1 in costs. For example, the 1 symbols in Bearer of Silence don't become C instead, since they appear in costs.
1 and similar is generic mana and is used for generic costs, payable with any kind of mana. You won't find generic mana symbols on spells and abilities that produce mana nowadays. C is specifically colorless mana. This mana can be produced by various objects and can be used to pay generic costs or specifically colorless costs.
Older cards printed prior to Oath of the Gatewatch, that produce colorless mana were printed with the generic 1 and similar symbols, but have been errataed to produce an amount of C. This was just a cosmetic change, not a functional one, the cards work the same as before, producing colorless mana. Colorless costs so far only exist on some Oath of the Gatewatch cards.
If you see the 1 symbol on an old card as a mana ability (AKA: Add to your mana pool for Kozilek’s channeler), that has been errata’d to the new colorless symbol (AKA: Kozilek’s channeler would now read: Tap: Add 2 )
If you see the 1 symbol in a cost, however, the two are not the same. in a cost cannot be paid with colored mana while in a cost can be. If they were the same, cards like warping wail that have both symbols in the cost would make no sense.