When should "Mom" and "Dad" be capitalized?

Solution 1:

"Dad" is a specific reference (when you say it you mean somebody different from when I say it), so it gets capitalized like any proper noun. On the other hand, "dad" is a common noun meaning "father" (anybody's). You only use disambiguators like "my" or "a" with common nouns ("my dad", but not "my Dad" just like you wouldn't say "my John Smith").

So the passage you quoted has it right. (3) is not capitalized because it's using a common noun ("a dad"), not a proper one ("Dad").

Solution 2:

When you are using the word "Dad" to refer to a specific person, it's standing in place of their name, and thus, like their name, would be capitalized. When you're talking about dads in general, it's a common noun.

Say you had a horse named Betsy and were re-writing the sentence to refer to her:

The one thing I learned from my horse was that it was good to earn the trust of one's children. Betsy has shown me how good that a horse can be. One day when I went to her house, and Betsy wasn't there, I ran outside....

Any time you're using "Dad" the same way you use "Betsy", it's capitalized. Any time you're using "dad" the same way you'd use "horse", it's not capitalized.