According to the Oxford Dictionaries:
There are three main uses of the colon:
between two main clauses in cases where the second clause explains or
follows from the first:
That is the secret of my extraordinary life: always do the unexpected.
It wasn’t easy: to begin with, I had to find the right house.
to introduce a list:
The price includes the following: travel to London, flight to Venice, hotel accommodation, and excursions.
The job calls for skills in the following areas: proofing, editing, and database administration.
before a quotation, and sometimes before direct speech:
The headline read: ‘Taxi Driver Battles Gangsters’.
They shouted: ‘Our families are starving! We need land!’
The main task of the semicolon is to mark a break that is stronger than a comma but not as final as a full stop. It’s used between two main clauses that balance each other and are too closely linked to be made into separate sentences, as in these two examples:
The road runs through a beautiful wooded valley; the railway line follows it.
An art director searched North Africa; I went to the Canary Islands.
As I see it, the Colon introduces what follows. In the two examples Oxford gives, always do the unexpected is introduced as that is the secret... and It wasn't easy introduces the next thing, which is what made it difficult.
For the semicolon, the two phrases are or are nearly equal in importance or significance. In first example, the road runs through it and the railway follows it are more or less equivalent. Yes, the railway follows the road, but one could just as easily say the railway follows the road and it would be fine. Meanwhile, the second example is even more clear: an art director went there and I went here. They're totally equivalent, grammatically.
In the examples you gave:
I gnaw on old car tires; it strengthens my jaw so I'll be better conditioned for bear combat.
Yes, the comment about strengthening his jaw does explain why he gnaws on tires, but it is an idea on its own: I gnaw on tires because it strengthens... I think a semicolon works better here because the focus isn't really on the strengthening more than the gnawing but on both more or less equally. Still, I wouldn't complain about a colon here.
When dinosaurs agree on something, they'll often high five one another; dinosaurs are all about high fives.
Here the second clause isn't introduced by the first, it's additional info about dinosaurs and high fives. I wouldn't use a colon here as the first clause doesn't introduce the second and the second doesn't really explain the first, it just adds to it. (P.S., Komodo dragons are the exception to this; never try to high five a Komodo. It pisses them off and they're already grumpy.)
Godzilla is a misunderstood creature; beneath his raging desire to set people on fire and eat them lies a gentle giant who just wants to cuddle.
This one, of them all, is perhaps the best candidate for a colon, but I think a semicolon works just fine too. Yes, the first clause could be an introduction to the second, but it could just as well be equal. Personally, I would go with the semicolon because I'm pretentious and want people to think I know how to use a semicolon correctly. (Pro tip: most people are very insecure around semicolons so they'll never correct you even if you're clearly wrong.)
Hope this helps!