Terminology in biology often gets a little fuzzy around the edges, sometimes as a result of overlapping definitions, or the desire to avoid overlap. For whatever reason, many eukaryotic pathogens are more commonly referred to as parasites. This seems to be more common with (though not exclusive to) multicellular parasites, like tapeworms, possibly leading to some definitions which exclude them as pathogens (even though they are clearly causing infectious disease). A more generalized technical term for pathogens, often used in biosafety protocols and research regulations is 'infectious agent.' This covers the whole gamut of infectious parasites, protozoa, fungi, bacteria, viruses, and non-living agents (like prions) with less potential for ambiguity.
But, the hard truth is that many biological terms lack a rigid and universally agreed upon definition. Conflicting or even ambiguous definitions can persist because such definitions aren't always necessary for biologists to perform productive research. I don't need to know if guinea worms are technically considered parasites or pathogens or both in order to study their transmission and pathology. I don't need to know if prions are living or not to study the infectious diseases they cause. What's more important is to have working definitions that don't cause confusion when used consistently in a given context (in your case, that context is learning year 11 biology).
In response to your second question:
Assuming that my teacher's definition is correct, why are most pathogens micro-organisms?
This answer is a bit of speculation, but, the vast majority of biological diversity on Earth is microscopic. Given how outmatched we multicellular organisms are, both in numbers and diversity, it's a wonder of nature that we persist at all. Eukaryotes are a small branch on the tree of life, and multicellular organisms are only a small minority of that branch. Plus, the most abundant and diverse biological entities on the planet, viruses, don't even have a permanent home on that tree.