Does the food I provide affect which kitties visit?

Offering more expensive cat foods, along with specific toys, will attract rare cats to pay a visit. More expensive cat food will also simply attract more cats in general.

The most obvious example of cats who are after food that costs money is Tubbs, who is only interested in food, and will only show up to eat (an entire bowl) of any food that costs money (fish).

Other rare cats are interested in playing with specific toys, and having expensive food around will increase your chances of coaxing them out to play with those toys.

I found information here and here in regards to this topic, and this also seems to be the consensus on Reddit.

Some rare cats will only show up when using specific foods or toys. You can find an incomplete list of which foods attract which cats on the Neko Atsume Wiki.