Yes, there's a difference between "a Celestial" and "a celestial creature".
"a Celestial": see the 3E MM, when it was a single entry. In 3.5, it's been split into "Archon", "Ghaele", "Angel", etc. This is a sentient inhabitant of the upper planes. They're not "humans who live on the upper planes" or anything, they're a distinct class of outsider. Note that word: Outsider. They're not humanoids, magical beasts, etc.
"a celestial creature": the analog of a mundane creature who's gained the celestial template by being from an upper plane. In 3E this was supposed to be for animals, vermin, etc., although in 3.5E it was clarified that you can have celestial humanoids (which are NOT the same as Celestials mentioned above). Creature type doesn't change, except that Animals or Vermin become Magical Beasts. They're not outsiders, though.
That is, in the first case "celestial" is a noun, in the second it's an adjective. It's the same as the separation between "fiend" and "fiendish".
To make things easier, read the 3E MM, Half-Celestial template, where it explicitly states that a Half-Celestial is the cross between a Celestial (which, like I said above, was a distinct entry in the non-template part of the MM) and an INTELLIGENT creature. They didn't include this flavor text in the 3.5E SRD, unfortunately, or else this'd be a lot clearer.
Also note that a Half-Celestial is an Outsider, so it can't be a cross between a celestial creature (which, at best, is a magical beast) and a mundane creature.
This has been clarified by WotC several times, but for some reason wasn't put into the FAQ.
So anyway, as to your story, the cross between a celestial riding dog and an awakened mundane dog? By the rules, it'd still just be a normal dog, not even intelligent. After all, the father's INT came only from being a Paladin's mount, not any sort of genetic thing, and the mother's INT came from a spell. MAYBE you could be generous and let the child get the celestial template too, but it wouldn't be any kind of Half-Celestial.
Even if you allowed it to be an intelligent creature, it would probably be like some sort of Planetouched dog. Sort of an Aasimar pet; smaller bonuses than the celestial template, for less ECL.