To clarify, there is no formaldehyde "vapor" when machining. Some, not all, MDF has formldehyde content in the glue that binds it all together, thus the dust it produces when cutting will contain formaldehyde. Most any dust (particlate) mask will filter well enough to catch the dust, the Achillies' heel being the seal from face to mask, which a good respirator does better. I have other opinions but I'll leave it at that.
The hepa filters in your vac will certainly capture the dust, the down side being the fine dust that MDF produces will tend to plug filter, so it may require frequent replacement. I can't see any way it could actually damage the vacuum. Cutting outside is an easy way to mitigate a lot of the concerns.
The Dyson's got a cyclonic first stage with a hepa final filter, right? Theoretically most of the dust should end up in cyclone bin or whatever it uses to contain the debris, but I have no direct experience.