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Reading closely, I would say Healing Spirit does not trigger the Unicorn Spirit. The Unicorn Spirit is triggered when you cast a spell that heals. Healing Spirit is a conjuration spell that summons a spirit. The exact wording about the heal is "...you can cause the spirit to restore 1d6 hit points...". It's not the spell, it's the spirit that's healing.
Think if you had a class feature whenever you cast a spell that turned someone invisible. Conjure Fey summoning creatures that can cast invisibility wouldn't trigger.
Yes, the description of healing is within the spell. But that's just because it's a one-off spirit that isn't listed anywhere else.