I added a couple of consoles that give +Deflector Dish, but I noticed that it isn't affecting the information in the Tyken's Rift tooltip. Am I misunderstanding how the stat bonuses works? I know +deflectors affects it, but why not +deflector dish

Starship Deflector Dish is a skill in the Starship Operations group. Items that give a bonus to it will increase your captain's skill-based stat for Starship Deflector Dish, which in turn will boost the magnitude or duration of any abilities affected by that skill.

Tyken's Rift doesn't appear to be affected by any skills (the ability description doesn't list any), so +Deflector Dish won't have any effect on it (and neither will +Deflectors, or any other skill-based stat bonus).

Hope that helps. You might find the Skills: The Numbers link in my sig to be useful, as well.