The Timescale License only covers the TimescaleDB code. Postgres code continues to be covered by the OSS PostgreSQL License [0].
So putting aside the question about API vs. psql level (again, the Timescale License was drafted to enabled this for "Value Added Services", e.g., where "such value-added products or services are not primarily database storage or operations products or services"), this license wouldn't apply for non-Timescale code.
[Timescale co-founder here]
[0] https://github.com/timescale/timescaledb/blob/master/NOTICE