Short answer, your hens don't have sq footage for their needs, and should be rehomed if more space can not be provided for them, regardless of your HOA rules.
Second, while you are welcome to make your argument, an HOA is a voluntary association of persons who have chosen to restrict one another's permitted land use to create a community of commonality. Don't like it? Don't move there. Nor is ignorance of the HOA rules a defense.
Looks like you are going to claim that "household pets" is somehow ill defined, and thus should permit your chickens. First, you are reading (and presenting) your HOA rules in isolation - a terrible way to read any legal document. Second, you are making that argument uphill... The preceeding line is that "NO ANIMALS OF ANY KIND (emphasis mine) may be raised, kept, or permitted" - this is legal-speak for "NO" to everything you might imagine, no definitions required. It then creates an exception for a reasonable number of "household pets" - setting aside that your chickens do not domicile in your home, and thus are not "household" in the way a dog, cat, fish, snake, or hamster might be - in the absence of a definition found in the HOA rules, the default is to look to the overarcing Zoning for either county or State.
Your
county defines chickens as "domesticated fowl" and domesticated fowl are part of "livestock".
Domestic animal means any animal typically kept as a pet, other than
livestock.
Livestock means ratites, psittacines, horses, mules, jackasses, cattle, llamas, alpacas, sheep, goats, swine, domesticated
fowl and any fur-bearing animal bred and maintained commercially or otherwise, within pens, cages and hutches.
Reading further, your county does allow the keeping of fowl for sale, subject to zoning restrictions and special permiting in zones
OTHER than your own, which appears to be R-5. Oddly, it seems you are allowed up to three pot bellied pigs...
Essentially, if you want to keep backyard chickens, move and build a larger home for your small flock.