Does Indian domestic law have a principal-role agency test?

Yes — Finance Act 2018 BEPS-ified Explanation 2(a) to s.9(1)(i): an agent who habitually plays the principal role leading to conclusion of contracts creates a business connection domestically. The treaties mostly do NOT contain that limb (only India–Netherlands, via the MLI), so a treaty-entitled enterprise takes the narrower treaty test via s.90(2) — the domestic/treaty gap is the whole point.