As you might have noticed when you were doing your dictionary research, the two can be used as synonyms, but horrible can also be used to mean unpleasant.
I had a horrible commute.
probably means the traffic was bad.
I had a horrific commute.
could mean there was a bad accident on the side of the road (or else I'm using a little hyperbole).
This is a subtle distinction in this instance.
In English, generally, you use the article "the" when referring to a unique entity in the world, one that is typically already known to the audience.
The prime minister is having a bad day.
If you're in a country with a prime minister, it's assumed that the prime minister is the one who leads your country. There is only one of them, and they are already known. In America (which has a president) you could not say that sentence without first mentioning what country the PM is from or which prime minister you mean.
In Britain, the prime minister is having a bad day.
or
We met the British prime minister. The prime minister was having a bad day.
But what about your example?
Although theoretically there can only be one second example, there could actually be more than one.
Second examples were sent to London and Paris.
As long as the animal sent to London is the second of that animal in London, and the animal sent to Paris is the second of that animal in Paris, there can be two second examples.
If you use "the second example", you are then saying that this is the one and only of that animal that is considered a second example in the context of this story. If you say "a second example", you are saying there could be other second examples in the context of the story, like a second example sent elsewhere.
Best Answer
This answer is going to assume that your question ultimately expands to be:
Answer: context.
Both the ordinal and the unit of time can be referred to using either article (the definite the or the indefinite a), so that's not going to help.
Commonly, when referring to the ordinal second, you'll see it describing something else:
Adding to the confusion, the ordinal can refer to "prize" or "place" without adding those additional words and still be valid English:
However, you'll commonly see ordinals with other ordinals:
When second is being used as a unit of time, they'll have a context relating to time:
Just to reiterate, the context is what will tell you one way or the other, and why writing grammatically correct English can be very important. Take the following terse sentence, do you know what they mean?
Without that context you just don't know!