They basically ask you to report it as a "memory jogger" for themselves so that they can do something when you call in (like clear you to land).
A standard airport traffic pattern is described in Chapter 7 of the Airplane Flying Handbook as:
The downwind leg is a course flown parallel to the landing runway, but
in a direction opposite to the intended landing direction. This leg
should be approximately 1/2 to 1 mile out from the landing runway, and
at the specified traffic pattern altitude. During this leg, the before
landing check should be completed and the landing gear extended if
retractable. Pattern altitude should be maintained until abeam the
approach end of the landing runway. At this point, power should be
reduced and a descent begun. The downwind leg continues past a point
abeam the approach end of the runway to a point approximately 45° from
the approach end of the runway, and a medium bank turn is made onto
the base leg.
It looks like this:
When the tower asks you to fly a two mile pattern, they want you to fly a pattern that has the downwind leg 2 miles out from the runway instead of the 1/2 to 1 mile that is normal. The base leg would still be turned when 45° from the approach end of the runway.
In this case, you are either:
- Doing pattern work in which case you fly the two mile wide pattern and should report turning base, or
- Approaching the airport and the controller would like you to enter the two mile wide pattern on the base leg. You should report the point where you would have normally turned onto the base leg if you were flying the entire pattern (45° to the runway). This would actually be both of your last two scenarios above: Established on the base leg, two miles prior to intercepting a two mile final (which for fun would actually be 2.83 miles from the runway).
