There is an artifact for each spec , 3 for most classes, 4 for druids and you can acquire them all and in an arbitrary order.

You get a quest to chose a weapon to pursue, then you pick one, do it, get the next quest to pick a different artifact, do that, rinse and repeat until you get the credit for acquiring them all.

I noticed some issues while playing around in Suramar on the PTR as my conditions where off.

G[40649]='K4096 L10 CD A:103832 T:113525 P:43979+43980 E21570' -- here the experiment is to require only the quest that selects the artifact and not the other quest that was just completed to be able to pick an artifact. hopefully this proves true as it means one could pick the artifacts in any order and all would work out properly.

The trick with these quests is simple, each time you pick one artifact, there is a hidden tracking quest that marks the spec for the artifact and how many artifacts are already chosen. So there are 3/4 different tracking quests for each spec.
So we can safely do "chose artifact1+spec a OR chose artifact 1 with spec b OR chose artifact 1 with spec c" "chose artifact 2, with spec a OR chose artifact 2 with spec b OR chose artifact 2 with spec c" in the conditions.

My slightly renamed quests in german for testing would be

G[43970]='Brewmaster Chosen - 2 Artefakt' -- to be confirmed
G[43971]='Windwalker Chosen - 2 Artefakt' -- to be confirmed
G[43972]='Mistweaver Chosen - 2 Artefakt' -- sicher
G[43973]='Zwei Pfade, zwei Waffen'
G[43974]='Brauhaus'
G[43975]='Rekrutierung der Schildmaiden'
G[43976]='Balance Chosen - 2 Artefakt' -- To be confirmed
G[43977]='Guardian Chosen - 2 Artefakt' -- To be confirmed
G[43978]='Feral Chosen - 2 Artefakt' -- To be confirmed
G[43979]='Resto Chosen - 2 Artefakt' -- sicher
G[43980]='Noch eine alte Waffe'
G[43981]='Affliction Chosen - 2 Artefakt' -- To be confirmed
G[43982]='Demonology Chosen - 2 Artefakt' -- sicher
G[43983]='Destruction Chosen - 2 Artefakt' -- sicher

The first draft will be coming up tomorrow with monks, warlocks and druids and the logfiles to provide insights into this.