I much prefer the name Apple TV.
ITV are very keen talk about their brand (ever since the regional commercial TV companies lost their identities and just became ITV) and particularly so now that they are launching pay to view stations. In the circumstances, it wouldn't surprise me if they cooked up this story and took it to the Mirror so they could talk about their "strong brand" in an article people will actually read because it is associated with today's hot ticket, Apple.
That sounds very plausible to me.