My definitions for the degrees of divergence are a little way down from the top in the second link. For the two you asked about:
Altered reality: history as we know it with overt (not hidden from the culture at large) magical or paranormal elements added (or some other non-magical element such as a species not living on our Earth that changes the setting from the world as we know it).
Small kingdoms that don’t exist in our history can place a story here if not in altered history. I would tend to put them here if the setting is an island and in altered history if they are in the middle of a continent, but others could argue differently.
Sorcery and Cecelia and the sequels The Grand Tour and The Mislaid Magician by Wrede & Stevermer and Mairelon the Magician and the sequel Magician’s Ward by Patricia Wrede. These stories are all Regency plus magic.
The Temeraire series by Naomi Novik (His Majesty’s Dragon, Throne of Jade, Black Powder War, Empire of Ivory, Victory of Eagles, Tongues of Serpents, Crucible of Gold, Blood of Tyrants, League of Dragons).
I categorize all of the Vanza/Zamar books by Amanda Quick here.
Alternate reality: stories that contradict in some way the currently accepted world-view of modern physicalistic science and world-view of the reader. This does not say that the reality they depict is in fact unreal or untrue, just that it is outside the limits of the current physicalistic paradigm. Alternate reality usually has its own very different history because of the different reality. Alternate reality often includes non-human intelligences such as Abominable Snowmen, Aesir, afreets/afrits/afrites/efreets/ifrits, angels, basilisks, banshees, Big Feet, brownies, cacodemons, cherubim, daemons, demigods, demons, devils, djins, dragons, dryads, dwarves, elementals, elves, faeries, fairies/fays, fauns, genies, ghosts, giants, gnomes, gods, goddesses, golems, Halflings, immortals, imps, incubi, jinni, kobolds, lamias, mermaids/men/folk, Minotaurs, naiads, nixies, nymphs, ogres, pixies, pookas/pucas, Redcaps, revenants, salamanders, sasquatch, satyrs, selkies/silkies, seraphim/seraphin, shapechangers, Sidhe, sprites, succubi, sylphs, talking animals, titans, Tritons, trolls, undead, undines, unicorns, vampires, wee folk, werewolves, wraiths, xerafeen, yeti, or zombies.
Psychic abilities (as distinct from working magical powers) can put a story in alternate reality if the reader doing the categorizing does not believe they exist in real life. If the reader accepts psychic abilities as normal, the same story would belong in one of the earlier categories.