OP, myFICO isn't owned by any of the big three (Experian, Equifax, or TransUnion). FICO works with all 3, in conjunction with lenders, but FICO doesn't own or is owned by any CRA. Maybe this is a bad analogy, but it's just like tires and cars. Goodyear doesn't own or is owned by any car manufacturer, they just do business together.
In fact, FICO competes with the three CRAs. The three CRAs sell their own score which you discovered. If you bought a three-in-one service via Equifax, then you don't have a FICO score. They gave you a rip-off score, in my opinion, that tries to compete with FICO. That's why your scores didn't match with your lenders.