How does the quality of the Fishing Rod affect the counter rate when fishing? If it doesn't, what does it do?

Question

What I mean by this is, say, will an Old Rod have you encounter a Pokemon less often, or, if not, what does the quality affect?

Answer 1

This should be true only for Generation I:

The Old Rod will always land a wild Pokémon and start a battle. The other two rods have a 1/2 chance of landing nothing each time the player uses it.

Starting from Generation III this was changed, and the only known way to improve fishing encounter is using an ability like Suction Cups or Sticky Hold:

If the first Pokémon in the player's party (even fainted) has the ability Suction Cups or Sticky Hold, the player is more likely to get a bite.
_{Due to an error in Pokémon Diamond and Pearl, the ability has no effect outside of battle. This behavior is kept in Pokémon Brilliant Diamond and Shining Pearl.}

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The only thing affected by the quality of your rod is the level of wild Pokemon. For example:

The Super Rod allows the player to encounter wild Pokémon that are at a higher level than those found using the Old Rod or Good Rod.