They combo with the Professional Training merit.

You've missed something. Professions give you Asset Skills; each profession has two, and you choose one that gives you the extra specialty. The merit Professional Training gives you more of them, and lets you do more with them.

• At Professional Training ••, you get a 3rd Asset skill. If this is character creation, you can pick your specialty from here.
• At Professional Training •••, Specialties in your Asset skills only cost 2xp.
• At Professional Training •••••, you can spend a WP to treat the next roll with that skill as a Rote Action (in some circumstances).

Connections and Lack Thereof

World of Darkness (Classic) and Chronicles of Darkness (formerly unofficially known as 'new' World of Darkness) are a spiritual predecessor/successor pair, not part of the same setting.

Changeling the Lost 2nd edition is not a 4th edition because is only connected with its namesake through the very broad theme of dealing with fae. They have radically different cosmologies, design philosophies etc.

The original World of Darkness was implicitly a shared setting, but each game line had some discrepancies that could not be fully reconciled with other game lines, particularly in terms of cosmology. Ultimately experienced GMs tend to prefer not even trying to make a single unified variant of the world. Also, the game mechanics of various Storyteller system flavours are not necessarily mutually compatible, even when comparing, say, VtM 1e and WtA 1e, or VtM Revised and MtA Revised. World of Darkness is famous or infamous (depending on who you listen to) for its metaplots and grand conspiracies.

The Chronicles of Darkness were originally published as World of Darkness by the company, which caused a degree of confusion. As a result, fans usually referred to it as the 'new' or less commonly 'rebooted' World of Darkness, despite the official branding not carrying such a qualifier. This naming collision caused a lot of problems, and the rebranding to CoD is one of the very few cases where I grudgingly accept that the Orwellian retroactive editing was warranted. It uses a unified system core, the Storytelling system, sold separately from all the 'splats' (game lines dedicated to specific types of supernatural entities). However, CoD as a setting tones down the metaplots and grandiose statements about the world, and is much more modular in design from the very beginning. That is, there's no reason to try figuring out the truth behind the contradiction between splat A and splat B because what the truth is even within a single splat is meant to vary drastically from GM to GM and from campaign to campaign.

Historically, the change between the two editions of nWoD/CoD was precipitated by the introduction of the God-Machine Chronicle. This highlighted a metaplot that was subtle in other sourcebooks but prominent in Demon the Descent.

Sorting the Game Lines

As for the flowchart or other scheme, here's a division of the main game lines:

World of Darkness (Classic), using an assortment of Storyteller system variants:

• Vampire the Masquerade (1st, 2nd, Revised, V20 and V5 editions). Elder vampires weave centuries-long plots, while their neonates are pawns.
• Mage the Ascension (1st, 2nd, Revised and M20 editions). Belief defines reality; fight to shape it!
• Werewolf the Apocalypse (1st, 2nd, Revised and W20 editions). Gaia is under attack; fight for nature and spirituality against corruption and pollution.
• Changeling the Dreaming (1st, 2nd, and C20 editions). You are a fae soul born to human parents and into a human body; defend the fairy-tales, magic and imagination against a stagnating world of grey banality. Fae and fairy-tale portrayal leans towards the positive.
• Wraith the Oblivion (1st, 2nd, Great War [a quasi-"2.5"] and Wr20 editions). You died; find your way to transcend into whatever follows the afterlife, while fighting against angst and bleakness, and against your own dark side.
• Demon the Fallen (Revised only). You are an Angel cast out by God from heaven into prison for rebelling with Lucifer. Now you escaped as a changed creature. Make your own cult and restore your glory in the shadows.

Other, 'minor' splats that fall under WoD, regardless of the edition, while exhibiting varied degrees of connection to one of the 'major' splats: Changing Breeds (various shapeshifters, such as the Ananasi werespiders), Demon Hunter X, Freak Legion (Fomori), Gypsies, Hengeyokai: Shapeshifters of the East, Hunter: The Reckoning, Hunters Hunted, Inquisition, Kindred of the East, Kindred of the Ebony Kingdom, Mummy (1st edition, 2nd edition, and Resurrection), Orpheus, Project Twilight (psionicists, hedge mages and investigators), Risen (wraiths possessing their corpses), Sorcerers.

Watch out: Mummy and Hunter have CoD analogues that can be partially distinguished by the subtitles (but still prone to confusion); some other splats might have those too!

In general, as of 2020-05-29, anything published as 20th or 5th edition is relatively new as far as WoD goes, i.e. it postdates the hiatus WoD took throughout approximately 2004-2010 (a period when CoD de facto reigned the book pipeline).

Chronicles of Darkness ('New'), using the Storytelling system base (1e or 2e):

• Vampire the Requiem (1st and 2nd edition, also known as Blood and Smoke). Like VtM, but elders are more on the senile side, and the metaplots have been drastically cut back.
• Mage the Awakening (1st and 2nd edition). Reality is not what it seems; find the secrets of Atlantis, and fight against those who try to keep humanity imprisoned in an illusion.
• Werewolf the Forsaken (1st and 2nd edition). Spiritual hunters in a more shades-of-grey world.
• Promethean the Created. (1st and 2nd edition). You're a brand-new being made from the detritus of the world, trying to become human in a world that hates or fears you.
• Changeling the Lost (1st and 2nd edition). You, a human mortal, got kidnapped by a fae, then escaped, but were changed by the experience; now you live a life of hiding or fighting against the fae's servants. Fae and fairy-tale portrayal leans towards the negative.
• Geist the Sin-Eaters (1st and 2nd edition; the least connected to its spiritual predecessor, and arguably not part of the 'main' list even). You died, but you got into a pact with an otherworldly entity and been given a second chance at life.
• Demon the Descent. You are a former angel that was dropped out of the God-Machine because you developed free will. And now the God-Machine wants you back as an unthinking piece.

Other, 'minor' splats that fall under CoD, regardless of the edition, while exhibiting varied degrees of connection to one of the 'major' splats: Hunter the Vigil, Mummy the Curse, Beast the Primordial, Deviant the Renegades. Watch out: Mummy the Curse has three books worth of WoD predecessors; Hunter apparently has about five!

In general, as of 2020-05-29, if you encounter something published as '2nd edition' recently, it probably belongs to Chronicles and had a 1e counterpart before the God-Machine Chronicle update hit the shelves (April 2013).

Bonus source of confusion: Monte Cook's World of Darkness is a very loosely related spiritual cousin which has some of the same creature types as Chronicles of Darkness, but handled differently, in a very different system, and in a setting where a big supernatural cataclysm happened recently and significantly changed the world from how we know it. The White Wolf community generally doesn't even talk about it.

BONUS Bonus book confusion: The book named A World of Darkness has no relation to the similarly-named World of Darkness book, nor is it a corebook. It is an oWoD book detailing a bunch of locations in the setting and focusing on vampires.