Advice needed on staining 10 year old fence
Ok let's talk outdoor staining.  You are right that the first thing you need to do is clean off any mold, dirt and green slime. As with decks, a good and cheap solution of TSP, bleach and water is a fine cleaner, no need for expensive premixed stuff.  Using a one gallon garden sprayer, apply your TSP/bleach solution. Let it sit a bit, but not dry. Scrub with a stiff brush and rinse with a pressure washer or high pressure hose. The drying time is obviously tricky in your area, but usually a couple of days of decent weather does a good job. You can somewhat judge the dryness by it's color as compared to the color when wet. 

Since the last stain you used was a urethane based acrylic, you really need to stay with an acrylic product this time. A pure oil based product may not sit well and raise the acrylic and create a mess. Also, since there was spar urethane in the last application, a stain only may not want to stick to the areas where the urethane is still in good condition. The urethane is a hard resin and will repel a simple stain. I would probably look towards using a similar product to what you used last time.

As far as what brand to buy, don't over think it. You are painting a fence, not a church! Avoid any bargain basement crap that is most likely low quality. I love Sekins, but it is very pricey. Most major brands are good. My advise is to try to use the same Behr you used last time. It will be formula compatible and the color will match.  Good luck