I really should have expected you to find a citation that would pull the rug out from under me! I yield. I always took that Weave of Magic sidebar to be some setting-specific info and ignored it. But you're right -- it applies to the entire D&D multiverse (shudder).Page 84 of the PHB "By 2nd level, you have learned to draw on divine magic through meditation and prayer to cast spells as a cleric does."
Druids and clerics change spells via prayer. Channel divinity to turn undead is prayer. Prayer is everywhere in the divine classes. And yes there are divine and arcane categories.
Page 205 of the PHB "The spells of wizards, warlocks, sorcerers, and bards are commonly called arcane magic. These spells rely on an understanding-learned or intuitive-of the workings of the Weave. The caster plucks directly at the strands of the Weave to create the desired effect. Eldritch knights and arcane tricksters also use arcane magic. The spells of clerics, druids, paladins, and rangers are called divine magic. These spellcasters' access to the Weave is mediated by divine power-gods, the divine forces of nature, or the sacred weight of a paladin's oath."
Despite that, I maintain that although clerics and druids prepare spells through prayer, the spells themselves are spells, not prayers.
If the intent was to limit arcana to lore about arcane spells, it would have been simple to incude that in the skill description, just as it would have been simple to include a bit about divine spells in the religion skill description.
And why does a druid have arcana as a class skill, anyway?
It's almost as if the D&D 5E rules are not internally consistent.