Rich,
If it's an Omega, and it's a Quartz, it's not a chronometer. Omega does not have its Quartz movements COSC certified; they're so accurate that it is really unnecessary (and would be an added expense to them and in the end to buyers). Most high end companies follow this same policy. Only certain of their mechanical watches are COSC certified. If accuracy is important to you, you can't beat a quartz watch.
As to activities: Obviously a watch costing hundreds or thousands of dollars, and designed to be worn while skin diving, should be safe to use while swimming or fishing. You should have the seals tested every year or two to assure that they are intact, and you should rinse your watch well with fresh water after immersing it in any body of water, especially salt water. The quartz watch is actually well suited for fly fishing (which requires repeated, sharp snapping of the wrist while casting) for the Omega quartz movements are supposedly certified to withstand up to 5000g's of force.
Many people shudder at the thought of actually wearing a $1000 watch in the water, but my feeling is if you don't want to get it wet, why buy a divers watch to begin with? I've worn Omegas for a total of about 7 years now; as a healthcare professional, I wash my hands sometimes 40 or 50 times a day, and I've never had a problem (knock wood). I am very careful about rinsing all soap off my watch, as I have read somewhere that some detergents, over a long period of time, can degrade the seals. Hope this helps.
