I think you may be confusing the function of two different coatings. A Protectant Coating can be single or multi-layer. MRC simply means that there are multiple layers of protectant. In Fact, B+W's F-Pro line can come with single or MRC. This coating protects against keys, rocks, angry celebrity's fingernails, hotwheels cars, or whatever other foreign object smacks the front of your filter. It does not necessarily protect against stray light. Protectant coatings can (but will not always) have the following additional properties: UV/IR cut or Oleophopic / Hydrophobic or Anti-Reflective, or other bespoke functions. Remember that coatings are actually tiny particles mixed into the glass, deposited as a thin film, or ion sputtered and are often many layers of such particles, even in the misnomer "single coating." This means that many functions can be performed by one coating, provided that the interference of layers is carefully controlled. High quality coatings are imperative to an imaging system and I encourage you to learn more.
So let's talk about the anti-reflection coating. A significant percentage of light striking an uncoated lens is reflected. In complex optical systems (read all modern camera lenses) this can become a very large amount. A/R coatings reduce reflection by up to four orders of magnitude (more than a dozen stops) and are therefore critical to eliminating flare and maintaining contrast. B+W filters have at least a single layer A/R coating on both sides. Some have the superior multi-layer interference coatings. I've not been able to find similar information on Haida filters. It is possible that the Haida filter does not have A/R on either side of the filter.
Brief discussion at a more technical level; The reflection of light as a result of incident light across a transition of refractive indexes is known as Fresnel Reflection. AR coatings work to minimize reflection by destructively interfering with the reflected rays. For optical surfaces in contact with air This is accomplished by creating a single coating with refractive index equal to the square root of the surface's refractive index [Ref pp137-142 Optical Systems Engineering (2011)]
Interference AR coatings have a distinctive red/green emission at highly oblique angles. It might be hard to spot on a 10-stop ND filter but if you look carefully you should be able to tell whether an A/R coating is present on either surface of the glass. Comparing it to your B+W should help you spot the coating.
What is the bottom line? If your Haida filter does, in fact, lack an A/R coating it is losing you up to 4% of your light and might be reducing your contrast by 2 to 4%.
I'm guessing the Haida filter was much cheaper than the alternatives so loosing 1/8 stop of exposure and contrast is probably acceptable. Note that when I say contrast I mean global contrast (dynamic range) and instantaneous contrast (resolving power.) Your optical system is also more susceptible to flare and glare than if an A/R coated filter were equipped but flare and glare are generally visible to the photographer and can be mitigated. I usually try to avoid any sort of product endorsement but I've personally tested B+W filters against the competition and there is a real difference. That said, in most situations you will only barely notice that difference.
A note on ND and light: A 10-stop filter is also known as an ND30 or ND3.0. It has an optical transmissive density (density) of 3.0. A 2-stop ND filter is ND06 or ND0.6. It has a density of 0.6. The 10-stop ND30 will allow just 1/2^10 or 1/1024th of the light to pass through. Despite this tremendous reduction in lumminance the quality of the light is not changed at all by a true neutral density filter. In reality the coatings mentioned above may have effects beyond the reduction in light. The question of whether a neutral density filter really is neutral involves a rant about production quality that I will save for another time.