Note that this only works for the first Eevee nicknamed this – otherwise the evolution remains random.

My experience says otherwise. I'd say that it's not as simple as saying the evolution remains simply random.

I evolved 13 Eevees and these are the results I got:

1. Flareon

2. Flareon

3. Jolteon

4. Jolteon

5. Flareon

6. Flareon

7. Vaporeon

8. Flareon

9. Vaporeon

10. Vaporeon

11. Vaporeon

12. Vaporeon

13. Jolteon

If we assume that each Eeveelution up to this point (Flareon, Jolteon, Vaporeon, Espeon, Umbreon) has an equal chance of coming up when you evolve Eevee then there should be a 40% (2 out of 5) chance that my Eevee evolves into Espeon or Umbreon. Unless I've done my math wrong there is over a 99.99% chance that at least one of those 13 attempts should have come up with Espeon or Umbreon. There has got to be some other factor at work: time of day, location, whatever. That, of the chance of getting an Espeon or Umbreon is less than the odds of getting a Flareon, Jolteon, or Vaporeon.

For clarification, I do have the latest update and have already caught several Johto Pokemon as well as evolved a Crobat so I know my game has Johto Pokemon available.