I suspect I may be showing my ignorance here, but aren't we talking about 2 differnet vortices?
If I dredge up what's left in my head from the wing design course, the trailing vortex is a function of the wing section, giving us the circulation term in 'Lift = rho * U * gamma" , and forms a line vortex roughly parallel with the trailling edge - and thus would exist for our infinite wing.
The tip vortex can be thought of as the result of the higher pressure air under the wing effectively spilling over the tip.
The interation between the two can get interesting, and is the bit that winglets can play around with.