That is an interesting summary and I like the "unconventional" approach of presenting post-medieval examples, too. This brings its own problems, though.

"Greensleeves", e.g., is from the Renaissance period which means that musica ficta was not notated, but had to be added by the performers according to some rules. These rules were in many cases ambiguous or fuzzy, but in the case of the SI in the first phrase, there was a clear rule: "Una nota super LA semper est canendum FA" (see this recent thread for an explanation of this rule). The version with the si bemolle is thus original, whereas replacing it with a si naturale as, e.g., in the popular orchestra arrangement by Vaughan Williams is a modern invention.

Moreover, it seems that the perception of the mood of the modes changed over time. Mode III, e.g., was used in medieval times for the "Exsultet" during Easter night and other exuberant stuff, whereas chorales of the reformation used it for sorrowful texts like "Aus tiefer Not" or "Erbarm dich mein".

@Chant_Supremacist I picked those up from various blog articles about the modern modes. I thought they'd be handy to include, as they might help readers to formulate their own thoughts about how to describe the modes.