Protons have more mass, and should be producing an inwards electric field lines.
First, the mass of the proton isn't important. The electric field is produced by the charge of the particle, not its mass. Protons and electrons have equal (but opposite) charge, despite their vast difference in mass.
Second, the direction of the electric field tells you which direction would the force on a positive charge be if you put it in the field. Since two protons repel each other, you know that the electric field from a proton must point away from it.
Also, If the above is true, the electrons would flow with the electric field instead of against it in a closed circuit.
An electron (which has negative charge) is attracted to a proton. That is, it experiences a force toward the proton. And the force on a negative charge in an electric field is in the opposite direction from the field vector. So again, this is consistent with the electric field produced by the proton pointing away from the proton, not inwards towards it.
None of this has any bearing on the difference between conventional current and electron current. Whether you work with conventional or electron current, you'll still consider the electric field pointing away from positive charges and towards negative charges. You'll just have a different convention for the direction you assign to the current relative to the movement of the carriers.