I prefer a topical membrane over a membrane buried between two layers of mud. Simply because it affords better control over moisture within the pan.

Here's the "old way", with the CPE or CPVC membrane buried between two layers of mud. Courtesy of Harry Dunbar.

My preslope is thicker than Harry's, a little over an inch at the drain then sloping up towards the walls. And I don't use latex admix in the mud.

Shifting gears to the "second" method:

With a topical membrane, you can use one sloped layer of deck mud. The membrane gets installed on the sloped mud. Then you tile on the membrane. A couple of types of topical membrane:

You can use Hydroban over sloped mud. If you can get a magic trowel like they use in this video, your installation will be a breeze.

Or Kerdi Membrane over sloped mud.

There are a few other topical membranes, but in general they are intended for use on walls only. To use them on a shower floor requires fiddling with the mud around the drain and the drain itself, etc. Hydroban and Kerdi fully warrant their materials and methods when used as shown above.

If I had to chose a "best way" I'd use Hydroban with the flanged drain.

If I needed "bulletproof vapor control for a steam shower" then I'd use Kerdi.

If you want the "best way to do a CPE/CPVC shower", follow Harry's methods.

One of my shower quotes is for the way it is shown in the "old way" link. In that link, it shows a step where the walls got cement board, but he doesn't show what went behind it. Is either building paper or plastic considered a "good" way to waterproof behind the cement board?

Yes, properly detailed tar paper or 6-mil poly is fine behind cement board.