There is a difference between "a fire-type pokemon" and "the fire-type pokemon".
In the indefinite example, "a", you are generally about your preference for fire-type pokemon. It isn't specific to the three pokemon that you are looking at, but a general fact. It means that if there was a different fire-type in the suitcase, you'd still choose the fire-type.
On the other hand by saying "the fire-type pokemon" you are saying that there is a specific example that you are speaking about, and you believe your listener will also know which specific pokemon you are referring to. And if there was a different fire-type pokemon in the suitcase, you might not choose it.
In the second example, the nickname is generic when it is first mentioned. But when it is mentioned the second time it specifically means "that nickname that you intend to give to the Charmander". As it is specific on the second mention, it uses "the".
So the question "is something specific?" is determined by the speaker, using their belief about what the listener will understand. If you think your listener will be able to answer the question "which?", then you can use "The"
I would choose a fire-type Pokémon from these three.
Listener won't be able to answer the question "Which pokemon" because I would choose any fire-type.
I would choose the fire-type Pokémon from these three.
The listener would be able to answer "which pokemon" since they can see that there in only one fire-type, and I am talking about it.
Would you like to give a nickname to the newly hatched Charmander?
The listener would not be able to answer "which nickname" since this is the first time I've mentioned nicknames.
Then what shall the new nickname be?
Listener would be able to answer "which nickname", it is the nickname of the Charmander that I just mentioned.