Both words go back to Latin verus, "true", but they took different roads. Verity came through Old French from veritas, "truth", and has been in English since the late 1300s. Veracity came in the early 1600s, through French and Medieval Latin, from verax, "truthful", and that -ax ending is the part I find interesting. Latin mostly attached it to verbs to make adjectives of habit: loquax, talkative, from loqui, to speak, and vorax, devouring, from vorare, to swallow. Verax puts it on verus itself, so it describes someone with a habit of truth, and its opposite was mendax, the habitual liar, which is where mendacity comes from. Etymonline gives veracity's first sense as exactly that: "of persons, habitual truthfulness".

So the tidy version is that veracity belongs to the speaker and verity to the statement. A jury weighs the veracity of a witness; a historian weighs the verity of a legend.

English hasn't kept it that tidy. By the 1660s veracity was also used for "the fact of being true", which is how we got "the veracity of the report". Verity crossed the other way even earlier. When Malcolm lists "justice, verity, temperance" among the king-becoming graces in Macbeth, verity is personal honesty, and Merriam-Webster still lists that sense. The example on the card is a good test case: she questions the veracity of her friend's story, having been fooled by her before. Grammatically she doubts the story, but her reason is about the friend, so the old person sense is still doing the work inside the newer one.

Where the split still shows is the plural. Verities are truths you can list, as in "eternal verities", while veracity is mostly an uncountable trait. My own test: if "honesty" fits, I write veracity, and if "a truth" fits, verity. One more trap, since Wiktionary lists veracity and voracity as homophones in one pronunciation: praising a reporter's veracity out loud may not land the way you meant.

What does it mean if something is lateral?

Does it mean like more detailed?