I would say that microwaving is neither.
Heating with conventional methods works through heat coming outside of the food. Conduction and radiation will heat solid foods immersed in a gas or a liquid. "Moist heat" means that the liquid is water, "dry heat" that you are using another fluid to transfer heat. The distinction is useful, because with water, you 1) can't get above 100 C (even in a steamer, the steam just condenses on the food), 2) you can hydrate starches and gums, and 3) taste-carrying components are "leached" into the water by dissolving. You don't get these with air or oil.
But a microwave does not use a fluid to transfer heat. It uses pure radiation, but not in the non-penetrating infrared range the way a fire does, but in the microwave range. So it technically does not function neither like moist nor like dry heat. And in practice, the implications of neither dry nor moist heat cooking are present when you have cooked something in a microwave.
It is like trying to decide if meat is fruit or vegetable. It is neither, and microwave cooking is neither dry nor moist. If you have a source which claims that "dry" vs "moist" is an exhaustive categorization of cooking methods, it is probably older than microwave ovens, or too elitist to consider them worthy of kitchen use.
Added your very useful comments to your answer
"I somehow have the feeling that you are giving the distinction dry vs. moist heat more importance than it deserves. It is a convenient shorthand for throwing a few common cooking techniques together, but not very precise. If you need precision in your expression, you should be using other terms, and probably not bundling techniques at all. But to answer your comment, it is not the content of the food that counts (so the fat and water inside the chicken don't matter). If you heat chicken on a hot teflon pan without oil, it is the pan which transfers heat to the chicken, so it is dry."
"The air in a broiling appliance is still relevant. But I said that in dry heat, you get heat from outside from both convection and infrared radiation, and in a broiling oven, you just have a higher infrared-to-convection ratio of heat sources. It is still dry heat just like any other oven. And still different from a microwave, where the radiation penetrates deep into the food before starting to warm it."