Wood – ‘glue and screw’ 1/2“ and 3/8” plywood sheets to substitute for 7/8″

Fasten your first layer plywood subfloor as you normally should.

The second (top) layer of plywood joints should not coincide with the joists and the second layer should not be fastened to the joists.

Article quote from - Position of Underlayment to Prevent Cracked Tile and Grout By Frank Woeste and Peter A. Nielsen For publication by the TILE LETTER Magazine (read whole article):

The 2003-2004 Tile Council of America’s (TCA) Handbook for Ceramic Tile Installation contains numerous details for a double layer wood floor system supporting ceramic tile. The thicknesses of the subfloor and underlayment are given in each case. Specific guidance on where to butt the underlayment end joints is not given for any detail. For example, for F142-03, the TCA Handbook states, “offset end and edge joints of the underlayment panels by at least two inches from the joints of subfloor panels; they should not coincide with framing below." It further states, “underlayment fasteners should not penetrate joists below." In the case of F150-03, the offsetting is not mentioned, but it does state, “underlayment fasteners should not penetrate joists below." The same holds true for F155; however, it also states, “face grain of plywood should run perpendicular to trusses, I-joists, or sawn lumber for maximum stiffness." The purpose of this article is to propose specific guidelines for the orientation and placement of underlayment, including end and edge joints, beyond the rules given in the TCA Handbook, to improve the performance of double layer wood systems. These guidelines are based on engineering science and field observations.

Also, since you will be using Ditra, make sure to stick with their guidelines for warranty purposes. Specifically see page 8 and 23 from The Ditra Handbook for your application.

I don't have any experience to say for sure one way or the other (except that I've not had any notable vibration problems with my front load HE washer on the first floor, which is just sheet vinyl on 3/4" plywood, near the end of the joists). But what I'd think you want is basically the same thing that that you want when you're soundproofing a floor; you don't want vibrations at the top to transmit elsewhere.

When soundproofing, what you really want is a sandwich of plywood, underlayment, plywood. So roughly what you're suggesting. Cork can work well, though rubber underlayments generally perform much better at the same thickness. You lose a lot of the benefit if you screw the top layer to the lower layer, though - you want the layers glued together, without anything hard bridging across the soft layer.