update: 6378.137 km is what I use now.
By convention the altitude of a spacecraft is the distance to the center of the Earth minus roughly 6378 kilometers, or some reference radius that is representative of the equatorial radius of the Earth. Spacecraft altitude is not really used as a precise description of a satellite's position, since its only a scalar and it requires a definition, but if you have to state some value for an altitude, use the distance to the Earth's center of mass minus some reference radius which is generally an average equatorial radius. For example I see the following equation a lot (e.g. 1, 2, 3, 4, 5, 6 and in this answer):
$$\text{altitude} = |\mathbf{r}| - 6378 \ \text{ or } \ 6378.1\text{ km}$$
Another way you can confirm this is to simply search this SE site for "6378". (it turns out most of the time those are from me, so it doesn't count). A more accurate value for the equatorial radius of the Earth could be used, such as 6378.137, but the difference is not meaningful because a spacecraft's altitude per se is not really a precisely defined quantity.
You can read more about the nadir and the question of precisely what point on Earth's surface is actually "directly beneath" the satellite at a given moment, and the difference between the geodetic subsatellite point rather than the geocentric subsatellite point in this excellent answer.
Now the last part of the question:
For the satellite with circular orbit which is constant - the distance between earth surface to satellite or the distance between earth center and satellite?
The short answer is "neither", but the answer that's likely to be closest to correct is that the distance from the satellite to the center of mass of the Earth is likely to be more constant than the distance to the surface, assuming you figure out whether you mean the geodetic or geocentric subsatellite point. This is because the the higher order gravitational terms beyond the monopole are weak and have only a small effect.
A quick way you can confirm that is to note that the Earth's radius varies by more than 20 kilometers from equatorial to polar, and yet satellites in circular orbits can have periapsis and apoapsis within a kilometer of each other. For example I chose two of the satellites in the "A-Train" constellation, which has a nearly polar orbit. Aqua and Aura and looked up their info in Wikipedia. Both of them have a difference between periapsis and apoapsis of only 2 kilometers or less.
A useful convention:
Using this soft definition, it is common to see orbits described using the altitudes at periapsis and apoapsis. For example, this discussion of a 625 x 625 km orbit of an Iridium satellite means that it's circular with a radius of 625 + 6378 km, and a Geostationary Transfer Orbit of 185 km x 35,786 km would have its apses also at about those values plus 6378 km added to each.