Definition of a field line? Ok so I finished by A-levels last year (english exams 18 year olds take) and we defined in my physics course we defined field lines (for an electric field) as: The path a free positive test charge would follow if acted upon no other force but the force due to the field itself. 
Reading more in to it more advanced text books (and the internet) seem to ignore this and not even mention positive charges or they paths they take. Saying that field lines are simply the direction of the electric field around the charge in question.
So here is my question, is the first definition (the path a free positive ...) correct? If it is not correct what path would a free positive charge take in an electric field. Thanks.
 A: 
The path a free positive test charge would follow if acted upon no other force but the force due to the field itself.

This is wrong. (Did you actually have a book that said, this? If so, what was the book? This would be a serious error.) A charge in free space will have an acceleration parallel to the field, but the acceleration is not typically in the direction that the charge will move. The velocity vector is in the direction of motion, not the acceleration vector.
An electric field line is what's known as an integral curve of the field. Given an initial point, the field line is the unique curve that passes through that point and whose tangent is always parallel to the field.

If it is not correct what path would a free positive charge take in an electric field.

If radiation is negligible, then to find its path you have to solve Newton's laws, just like you would for any other force. For example, say we have a uniform electric field pointing down, and a positive charge is initially moving in a certain direction. Then the form of the problem is identical to the problem of a projectile moving in a uniform gravitational field, and the particle will move along a parabola with constant acceleration.
From a more advanced mathematical point of view, there are actually some reasons to prefer different graphical representations than the traditional one involving field lines. There is a nice visual treatment of this in Warnick, "Teaching electromagnetic field theory using differential forms," IEEE Transactions on Education, 40 (1997) 53. The paper can be found online (probably illegally, depending on the laws where you live). For example, the electric field can be visualized using a sheaf of equipotential surfaces, which a mathematician would consider to be a representation of something called a one-form. There is also a representation using flux tubes.
