Ed
I hesitate to recommend a person go or not go to a area I enjoy because of the vast difference in ideas how to relax & enjoy ones self. But what the hell, it's not like your going to show up at my front door & kick my ass
Having said that--We went by water taxi from C/C to A/C for lunch & afternoon tour of the island 2 winters ago, broke our heart to see what had become of the once quiet peaceful island. The photos I saw on the site you posted appeared to be from the old days, especially the shot of the main street. Now the street is filled with trucks, cars & very congested & dusty.
Touring the island will use up 1 day then you have to start traveling off island if vegging out @ the beach is not for you. Belize is a beautiful lush country & very enjoyable touring the mainland if your into that. Lot's of caving, river rafting etc. but you can read up on all that.
Simply put if you want quiet, laid back & vegging, Caye Caulker, which by the way has very little in the way of sandy beaches. The "split is the popular gathering place for sun & swim. Open air dinning with locales & tourists is most enjoyable. You can do all the same trips to the same places as you can from A/C It is my favorite, I have made friends there that I miss dearly when I'm not there.
Ambergris-- more restaurants, stores, shops, tour operators, resorts etc. much busier, noisier, bigger island.
They are located very near each other so you can take the water taxi & visit in about 15 min. The water taxis are a hoot & would recommend (depending on your arrival) to take a water taxi to your destination from the mainland, that way you get a cheap tour of the outer city & out islands. If you sit next to a local that speaks English (some don't) they are happy to give you a running commentary.
Regarding safety--I believe Belize ( excluding the city) is safer than most, but I would always recommend tempering that with a large dose of common sense.
The short answer I guess is I would not enjoy A/C if I went there for a first holiday in Belize.
I will add that our first visit to Belize we spent the first 2 weeks touring from the north end to the south end of the mainland in a rented car & absolutely enjoyed every moment.
Craig