Paint Exterior Door that is in Sun
I take it for $700 you are paying people to do this work.  And I gather these are garden variety handymen using random products bought at the local builder supply; when people say "polyurethane" without any qualifiers, they usually mean the Home Depot stuff.  

Have you talked to marine/boat painting places?  There are a variety of products intended for marine brightwork which will perform much, much better.  Because, obviously, they operate in VERY challenging environments. 

Oh, and to head off a common misnomer about marine products, "Spar" varnish is not the stuff. Spar varnish is lousy varnish which has all its best properties compromised away so it can flex without cracking, as it needs to do on spars.  If you're not regularly bowing that door 2" every time you open it, you don't need or want spar varnish. 

Now if you're really, really hellbound and determined to paint that door, the products your painter was talking about are alkyd enamels.  They're not water-based, so they are stinky.  They are also illegal to apply to buildings in some jurisdictions, but there's a weird loophole that lets you buy them in quart cans.  My Sherwin Williams dealer cheerfully sells me alkyds by the gallon, but is always "out of gallon cans" and sends me out the door with four quarts.  (I actually am painting industrial machinery; I think they just don't believe me). 

While you're at the boat place, you can also ask them about marine LPU.  This is a 2-component linear aliphatic polyurethane like you haven't met before; same basic stuff as Imron and what they paint airplanes with.  That stuff, I do use on architectural surfaces.  I have doors I painted 8 years ago. You can't tell I didn't paint it last week.