Huh? I dont think you are comparing apples to apples there. "Finishing" a beer with oak cubes sounds like adding them once the fermentation has been complete for a subtle oak character. "Aged" sounds more like a description of the cubes themselves maybe? Like type of wine/liquor they used to hold? Or possibly aging a beer for a long time in oak barrels for a prevalent oaked character. Then "fermentation" sounds like you are using it to impart brett/bacteria that is contained into the oak cubes into the beer for a secondary wild fermentation
As far as I know, none of those are "types" of oak cubes. Thats like French, American, dark, light, etc