Ok, here's my take.

First: If you can afford it a 2 speed filter is ALWAYS the way to go. Mine has a simple Hi-Lo switch on the back of the motor. You can ALWAYS run the pump at full speed should you not be happy with lo. But lo is quieter and cheaper.

Second: PBs and pool salesmen LOVE to sell oversized pumps. Go to PoolSolutions.com and read the tip on "The bigger Pump scam". I have 20,000 gal rectangular A/G with a deep end and I use a 1 hp Hayward SuperPump 2 speed--with solar panel, and run it at low speed 95% of the time. Other brands make 3/4 hp 2 speeds that spec out just as good as the Hayward.

Third: a properly sized and maintained filter of ANY type can keep your water clear. PBs love carts--I think they must be high margin, they are an easy install, and for most of the first season they don't need to be cleaned. IF you can figure out how to clean them without making it a half-day soaking-wet job you may be delighted with them.

For DE, they do a WONDERFUL job, but I suspect the consensus here is that an ordinary one, sized properly, will do just as well as a gadgety one, and will be far cheaper to maintain.

You weren't even interested in sand but sand can also perform far better than they lead you to believe. If your sand filter is slightly oversized (mine is) at low speed on your pump in cleans better. As the sand gets older and dirtier, it also filters better. Finally, adding a little DE powder after you back wash will down-right POLISH your water--and I've gone 2 seasons already on an $8. box of DE. You cannot add DE to a cartridge. (there is a fiber substitute for DE that you can use in a cartridge).

You should also use "skimmer socks" in your skimmer with any filter. They go in like a Mr. Coffee filter and hook over it. I find they last about a month and reduce the load on the filter by stopping stuff at the skimmer. Skimmer socks are usually about 3 for $5 or 4 for $10 depending on the brand and the store. Some people use old panty hose but I suspect it doesn't filter nearly as well.

But I think the advice you've gotten here on 2 speed pumps and pump sizing before I chimed in was just right on the money. It's your pool, not the pool builder's.

Since it looks like a concrete/shotcrete/tile/plaster pool, you may want to use Tri-Chlor pucks early on. They add acid and stabilizer(CYA) which you will need. When CYA hits your target level (generally max 50ppm) you'll stop using the pucks.