Here are some good points against using plywood, the source is talking about tiling a floor but most of the points are valid for walls as well.
In my opinion it's better to go the "overkill" route, then have to redo the job in the future because something didn't turn out right.
The cost difference between plywood and cement board is negligible, so your not going to save any money using plywood (unless you already have it).
Do the job right and use cement board, when you don't have any problems and you can enjoy the tile for years to come you will never regret the decision.
Perhaps more important than the size of the screws (or other fasteners) is what they actually grasp. With any significant load, a screw will not hold by itself in tile or plaster. The tile will chip and the plaster will crumble.
One possibility is to use an anchor through the tile and plaster that expands laterally. There are several types (plastic, metal, etc.), but these are also generally used for lighter loads that are close to the wall, not a shelf that sits on brackets that act like levers, tending to pull them through the wall:
![plastic anchor](https://i.stack.imgur.com/A6Iu0.jpg)
If your plaster is attached directly to masonry, there is an expanding anchor that is meant for that:
![lag anchor](https://i.stack.imgur.com/OvqJW.jpg)
It must be placed in the masonry, not the plaster.
If there is a hollow space behind the plaster, a third type of anchor that mushrooms behind the surface could be used:
![mushroom anchor](https://i.stack.imgur.com/eE9Oq.jpg)
A fourth kind is a toggle that is inserted folded, but then opens behind the surface of the plaster and its fastening:
![toggle1](https://i.stack.imgur.com/1ED3J.jpg)
![toggle2](https://i.stack.imgur.com/i11Kk.jpg)
If the wall behind the plaster is hollow, this may be the best type of fastener.
If there are studs behind the plaster, the strongest fastening is to align the brackets with the studs and go through both the backsplash, plaster and finally into the studs with wood screws (wooden studs) or screws designed for metal studs.
Best Answer
Taking a look at their web site, it appears that "Wedi board" is, effectively, a foam version of drywall appropriate for both wet & dry locations. (I looked it up because I had no clue what it was. I included the info here because I figure others in the US may not know either.)
Their web site shows the washers installed with a Phillips head screw through the center hole which appears to be recessed slightly from the surface.
A pan-head screw would work nicely because it's got a flat bottom to hold nicely in the flat of the washer's recess, however the domed top may stick up above the recess and create a bump in the finished surface. At a minimum, your tile mastic (making assumptions...) would have to be thick enough to cover the screw head. If you can find a pan-head with a shallow dome available with a waterproofing treatment, that may be the way to go.
A typical wood screw has a sloped shoulder that will crush into the washer's recess a little bit, but has a flat top that you should be able to get flush with the surface of the washer.
I would think that pretty much any wood screw would work in that situation.