The significance of personality & power level

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.

No, cats bring gifts depend on the items you placed, the food and the most important, the time their stay on your yard. The time could be more or less by the food or the items. Also, there are specific cats that only appear with a certain items + food. i.e Tubbs only appear if you have food that is not the "free food", he will come and eat the entire bowl and stay for a short time.

You can do a test, puts some food, see what cats come first, wait atleast 20+ minutes more and then see what cats came after the first one. Next quit all items. Doing this, all the cats leaves and you can see how amount of fish the cats give you.