Yes bug and dog describe the same thing, the (usually) static logo found in the corner of the screen. The beeb always like to come up with a different term for stuff. What the rest of the world calls Chromakey, the BBC calls it CSO (Colour Separation Overlay).

I assume the DOG term comes from a time when on screen graphics were primarily generated using rostrum cameras and a combination/selection of chroma and luma keying. Generating anything 'digitally' would be pretty cutting edge.