As you are saying, the magnetic field strength for the Earth is at its surface (that is normally available without digging), that is the closest available connection between the Earth and the magnet/metal.
The other magnet's strength is the same way, a refrigerator magnet's strength is meant by attaching it to the surface of the refrigerator, thus creating the closest possible connection between the magnet and the metallic surface of the refrigerator (without damaging the refrigerator or the magnet).
Now your question is, how can a refrigerator magnet be stronger then the Earth's magnetic field? The EM force of the Earth is different from the gravitational field. In case of the gravitational field, the Earth stress-energy (not mass directly) is very much important, and in some ways, the huge mass (in reality stress-energy) of the Earth makes you feel it should have a very strong gravitational field.
Now with the EM field, it is very different. It does not matter in reality how huge a mass (or stress-energy) the Earth has, even a small sized magnet can be (and in your example it is) stronger then the Earth's magnetic field. It is an intrinsic property of the EM force, where it can be canceled out inside the body by opposite charges. That is what happens, thought the Earth is huge, it is full of opposite charges that cancel out, and you only feel the net force (which is not even as strong as a small magnet).
Now with the magnetic field it is a little bit more complicated, because the Earth's magnetic field is due to a liquid outer core, that moves and creates electric currents. The rotation of the Earth on its axes causes these currents to create a magnetic field.
Now the magnetic field strength drops with distance very quickly, and since you cannot touch the liquid outer core inside the Earth, you are far away from the actual magnet of the Earth itself when you are on the surface.
When you attach the magnet to the refrigerator, that is as close at it gets to the magnetic field of the refrigerator magnet. But the refrigerator magnet is far away from the Earth's liquid core that generates the magnetic field of the Earth.
Your experiment would have a different result if you tried it very close to the liquid core (disregarding other effects like heat etc).