It will depend. If a star system as more equal amounts of oxygen and carbon, we could have planets formed not only by carbon or diamonds, but mainly of carbides of various materials, and having a planet made more of iron and carbides than diamonds. Such planet may have more oxygen. Maybe not water oceans, but why not oceans made of hydrocarbons or even ammonia? More similar to Titan, maybe at higher temperatures. Even if lack oxygen, we could even have silicon combining into carbon.
So maybe not all carbon planets are diamond planets, even if they have diamonds... Such as Earth has.
To me, carbon planets are perfect to form Titan-like worlds, even more hospitable to life than the icy Titan. It could have a mantle rich in hydrocarbons and deep seas of methane and ethane... Imagine that! And for hotter carbon planets, maybe seas of oil rich substance. So for cold carbon planets, hydrocarbon environment and for hot carbon planets, oil rich environment. Maybe even seas of asphalt!