Plane changes are expensive in terms of fuel consumption, making it essential that any launch intended for rendezvous with the ISS must target an orbit whose orbital plane is very close to that of the ISS. This requires launching a vehicle in roughly the right direction, and at roughly the right time. The question mentions that this addresses only two of the orbital elements. What about the remaining elements?
There are several rules regarding rendezvous with the ISS. One is that even if the visiting vehicle goes completely dead at exactly the wrong time, it will not come close to the ISS within an extended period of time. Another is that the visiting vehicle needs to take advantage of the fact that the ISS is a cooperative target; the ISS has multiple navigation aids explicitly intended to make rendezvous easier. The combination of these two means that rendezvous is a slow process. The fastest rendezvous takes six hours from launch to rendezvous; some vehicles take several days.
I'll discuss the concept of phase angle before I answer the question. The ISS is in such a nearly circular orbit that argument of perigee and true anomaly aren't that well defined. What is well defined is the argument of latitude, the sum of argument of perigee and true anomaly. The phase angle between two spacecraft is the difference between the those spacecrafts' arguments of latitude.
The answer to the question depends very much on the rendezvous strategy used by the vehicle's mission planners. Some of the issues include
- Whether the vehicle launch window is designed to allow for fixable issues. SpaceX and Russia use nearly instantaneous launch windows (one second launch windows). Others have more flexible launch windows. A delay of just one second means the ISS is 7.7 kilometers ahead of where it would have been had the launch occurred at exactly the planned time.
- Whether the amount of time that passes between launch and docking/berthing is a few hours or several days. Some vehicles take several days, others, just six hours. Taking several days adds a lot of flexibility, but at the expense of taking several days.
- Whether phasing orbits used after orbit insertion have semi major axes that are greater than or less than that of the ISS. Phasing orbits that are above the ISS need to have the vehicle be ahead of the ISS so that orbital mechanics will eventually bring the vehicle closer to the ISS. The reverse is true for phasing orbits below the ISS; phasing from below requires the vehicle to be behind the ISS.
- Whether the final few orbits have semi major axes that are slightly greater than or slightly less than that of the ISS. The same basic concepts for phasing orbits apply, but the phase shift needs to be much smaller for those final few orbits.
I'll look first at the many orbits (several days) approach to rendezvous. These approaches use phasing orbits whose orbital period is slightly larger (approach from above) or slightly smaller (approach from below) than that of the ISS so as to eventually reduce the phase difference. The launch targets matching the orbital plane, but has a non-zero phase difference, the sign of which depends on whether the approach is from above or from below. A properly designed rendezvous strategy reduces the magnitude of the phase angle with each orbit.
With this many orbits strategy, changes in the spacecraft's orbit are initially entirely up to mission planners, and in real time, to mission controllers on Earth. Those mission controllers know the orbits of the ISS and of the visiting vehicle. The goal is to eventually bring the visiting vehicle to the point where it is somewhat close to the ISS and in somewhat the same orbit as the ISS. The approach to far field must keep the visiting vehicle from having any reasonable chance of colliding with the ISS. The transition from far field to near field rendezvous marks the point at which the visiting vehicle can finally see and communicate with the ISS. Visiting vehicle navigation transitions from absolute navigation to relative navigation at this point.
The visiting vehicle will be slightly behind the ISS at this transition point if the last few orbits follow an approach from below strategy, slightly ahead if it follows an approach from above strategy. Regardless of strategy, each orbit will take the visiting vehicle closer to the ISS. Eventually the visiting vehicle will be close enough to perform its final steps; how long is vehicle-specific.
What about faster rendezvous designs? Those designs must necessarily eliminate many of the intermediate steps and must drastically reduce steps from far field to near field to final approach. But even these fast rendezvous designs are a bit conservative. Unlike sci-fi, where rendezvous happens shortly after launch, even the fastest of rendezvous require multiple orbits between launch and rendezvous.