In my experience "appropriate/proper swim attire" simply means that it is at the discretion of the establishment to ask anyone to change their clothes. They don't post specific rules, because they don't want to get into semantic discussions. If briefs are allowed, can someone wear a see-through brief? They'd rather not get into that line of reasoning.
For practical purposes, these are the rules that I think apply in 99% of situations. If an establishment has a different policy, they will usually state it explicitly.
Men:
Lined briefs, square cuts, jammers, swim trunks, and board shorts are all appropriate. Briefs should have a full back coverage. Sides should be 1" or greater. Rash guards may be worn.
Women:
Lined two-piece or one-piece swimsuits are appropriate. All swimsuits should have moderate (3/4) back coverage. Board shorts and rash guards may be worn over swimsuits.
I've never heard of anyone having issues when following the preceding rules. If an establishment does not explicitly state that briefs aren't allowed, they should be acceptable.
Beaches
Beaches tend to be far less restrictive and usually guided by state/city decency laws. Some states allow women to be topless or persons of both gender to be nude.
In Seattle nudity is allowed as long as it isn't indecent exposure. A number of waterfront city parks are frequented by nudists without issue. When the rules aren't posted, the genitals and (women's) breasts must be covered. In more conservative areas, the buttocks should be covered too. If you're not comfortable walking back to you car in a wet suit while a cop is watching, it's probably not appropriate.
Women's Back Coverage
Back coverage on women's suits tends to be the only big gray area. Because even modest one-piece competitive suits leave at least 1/4 of the buttocks exposed, places tend to be more lenient when it comes to how much back coverage is required. Often, as long as it's not a g-string it's okay. In general, 1/2 coverage will be considered appropriate anywhere and 1/4 coverage will be fine on most beaches.
This would be fine almost anywhere.
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However, this bottom might be a problem at certain establishments. In most cases, however, it would be acceptable on the beach.
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