"Bad with something" or "bad at something"?

In a question on Spanish.StackExchange, a question came up about expressing that you are bad at remembering or doing something. Is one "bad at something" or "bad with something" (nouns)? What about "bad at doing something" vs. "bad with doing something" (verbs)? If there are regional differences, what are they?

Examples:

• I'm bad with names.
• I'm bad at names.
• I'm bad with cooking.
• I'm bad at cooking.

Solution 1:

My initial thought is that bad at (or good at) is followed by a word that describes some kind of activity and that bad with (or good with), is followed by the name of a thing or a type of person. It works with at least the following:

Bad at: swimming, math, lying, football, acting, planning, French, cooking, chess

Bad with: children, his hands, money, computers, foreigners, figures, animals

Solution 2:

These are very similar. Good/bad with shades towards naturalness and affinity, or lack thereof. A person who is good/bad with computers is able to relate to them comfortably. Good/bad at shades towards skill or competence, or lack thereof. A person who is good/bad at computers is able to operate them competently.

Consider that a person who is bad with children might not relate to them well, but a person who is bad at children might have difficulty conceiving.