Noon vs midnight:

Since "A.M." and "P.M." mean "Ante Meridien" and "Post Meridian" (Latin for "before noon" and "after noon"), neither "12:00 a.m." nor "12:00 p.m." makes any sense, from a strictly logical standpoint. (Twelve hours before noon and twelve hours after noon are both midnight; neither would mean "noon.")

That's why I am always confused when someone writes or prints any reference to 12 am or pm ~ they should simply say "noon" or "midnight" to be clearly understood.

However:

12:01 am is quite obviously a "morning" (ante meridien) time ~ wee-wee-hours of the morning, of course ~ just as 12:01 pm is one minute into the early efternoon. By extension, therefore, if someone is inconsiderate enough to confuse matters by referring to straight-up twelve o'clock as "AM or "PM," then 12 am should probably be understood as referring to midnight (because one minute later, 12:01 am, is a morning hour) while 12 pm would by the same logic be noon (again, be extension from recognizing 12:01 pm as one minute past noon).

I think it bass-ackward to blame this confusion on the analog clock; it is clearly a product of the digital age. Before the proliferation of digital timepieces, people almost always automatically said and wrote "noon" and "midnight" as a matter of course, and were much less likely to feel the compulsion to express these times as "12:00" and then be confronted with the confusing task of deciding whether to call it 12 am or 12 pm.