The point of confusion, it seems, is with the electric field inside the conductor. The electric field in the conductor is only a result of the fact that the the conductor has some resistivity (a finite conductivity). To clear this up, consider a perfect conductor. Current flowing in a perfect conductor does not require an electric field in the wire. As long as the current remains at a constant rate, the current will flow even without an electric field in the wire. The analogous situation with physical objects is of course an object floating in space at some velocity. Nevermind how it got to that velocity for now, the fact that it is moving at a constant velocity relative to some observer is the important part. It requires no force to keep it moving.
In a typical conductor, there is resistivity. The same analogy with moving things applies: your object is now floating in a gas, moving through say, air. The object will eventually stop due to air resistance, though the effect is very small. As well, if you apply some constant force to the object, it will accelerate to some velocity, and then it will hit terminal velocity where the air resistance is equal to the force applied. This is what happens in a wire that has a voltage across it, though note that this is a short circuit situation, with very high current levels.
In a perfect conductor, there will also be no voltage drop across the wire, even when current flows through it. The only things that can stop current flow in a perfect conductor with a voltage across it is the source of the current, which will probably explode from trying to supply infinite current. In a perfect conductor, with a voltage applied to it, current will just keep climbing up and up, like an object in space that has a constant force on it will keep trying to reach infinite speed.
It may seem like I'm rambling on, but this description of ideal and non-ideal currents relates directly to the equations you present. The first equation says that current is directly related to the applied electric field, and some quantity of charge that will flow. Note, this charge is not the surface charge on the conductor, which you mentioned. We know from above that the electric field in an ideal conductor will cause infinite current, and only the source of the current can stop it. In fact, that charge in the first equation is the source of the current and it is limiting the current flow. If you alter the electric field, now, the amount of current will vary because there is a force on a limited but constant supply of charges.
That's a bizarre and uncommon situation, in that form. I would wager that this equation is used to solve for the other quantities. For example when you know that a current J flows through some material with conductivity S and you want to find the electric field in the material.
The second equation relates current to the drift velocity of the electrons. This assumes that you know what the current is, and that it is not changing. Then the drift velocity can be determined simply using that equation.
So you see, these equations are using different variables, and thus they should not be taken together. The first equation describes current flow in terms of a charge moving due to an electric field. The second equation describes current flow in a non-ideal conductor and how it relates to the speed of the electrons.
For more information, look into conduction mechanisms, as there are several models that attempt to describe why objects have a specific conductivity.
Hope this was helpful. Feel free to comment questions and unclear points.
Sam
