You can use a latex primer and I often do for projects like this because they are easier to use and do not stink, but be prepared for some stains that may bleed through the latex primers. You may need to spot prime those stains with an oil based primer. I usually have a few spray cans of Kilz on hand for this.

I would use BIN because it sticks better than other primers and will kill stains. It's shellac based (alcohol) and pretty smelly. If you can't use that, then you can try an acrylic primer but there may be stains that bleed right through that. Even if you go and buy some 'stain killing' acrylic primer (most manufacturers make one) it is likely that it won't really kill stains and you will have to use Kilz or BIN.

Durable in what way? Oils are generally more "Mar" resistant as they provide a harder coat. They then don't last as long by way of fade and rust, especially around a pool. Alkyd means oil based to me. Where are you seeing water based product? Acrylic usually means water based. Make sure what you already painted is adhering well and paint away!

When they say don't "cheap out" LISTEN!! Take my advice, I learned the hard way. You can a) paint one coat of cheap-y paint, and then re-paint it year after year, and re-scrape, and get a sunburn and waste weeks on end, or.... b) pay the money up front, and do it right. Expensive lesson, especially when you are painting a huge garage. DON'T use those vinyl spackling buckets that cost 2.79. Get good exterior caulking, whichever one costs the MOST is probably the best one. Be patient, just let it dry all the way. Scrape off EVERY peel, no matter how long it takes. If it's REAL bad? Get the spray on stuff, and get every last bit off. That's what I wish I'd done. And I did scrape to bare wood, but used crummy spackle and crummy paint, and one coat. Saving $50 on paint and materials is now probably going to cost me 16 hours time, and over $150 to fix. If I could afford it, I would spend $2500 to side the thing and never do this again. Spend the money up front, take your time. A GOOD paint job will last 10-20 years. Mine lasted 16 months, and looks worse than when I started. And I had professional help. Who told me to buy the good stuff, and I ignored him, thinking how smart I was. And don't try to paint outside when it's 80 degrees. ACHHHHHGGG!