How to join a traffic pattern when passing overhead?
What is the correct way?
Is there a difference between controlled and uncontrolled airports? What about the procedure when approaching from other angles?

Well I'll address your second quesition first: There is ABSOLUTELY a difference between controlled and uncontrolled fields. At a controlled field you do what ATC tells you to do (and if there's any ambiguity you ask them to clarify what they want you to do).

At uncontrolled fields in the USA the standard way to enter the VFR traffic pattern is a 45-degree intercept to the downwind leg. The FAA's pronouncements on the issue can be found in AC 90-66A.
A brief and relevant excerpt:

a. Prior to entering the traffic pattern at an airport without an operating control tower, aircraft should avoid the flow of traffic until established on the entry leg. For example, wind and landing direction indicators can be checked while at an altitude above the traffic pattern. When the proper traffic pattern direction has been determined, the pilot should then proceed to a point well clear of the pattern before descending to the pattern altitude.
b, Arriving aircraft should be at the appropriate traffic pattern altitude before entering the traffic pattern. Entry to the downwind leg should be at a 45-degree angle abeam the midpoint of the runway.

(Emphasis Added)
You will find a rich variety of opinions (a polite way of saying &quot;hidebound arguments&quot;) among pilots on the correctness of various other entries, but the 45-to-downwind is what's expected by most pilots in the US, and it is generally the safest way to enter the pattern (you will be where everyone else expects you to be, and you will have plenty of time to look at the pattern and figure out how you're going to sequence into it).
The 45-degree intercept assumes that you are on the side of the runway where you will be flying your downwind leg. If you are on the opposite side you need to position yourself to maneuver for a 45-degree intercept. I am personally a proponent of the technique illustrated below:


Cross midfield 500-1000 feet above pattern altitude.

At airports where there are multiple pattern altitudes specified, such as a &quot;jet pattern&quot; above the light aircraft pattern, cross at least 500 feet above the highest pattern altitude to ensure traffic separation.


Proceed away from the airport and clear of the traffic pattern before descending.
Make a descending turn to intercept the 45-degree entry

Ideally time your turn and descent such that you're at pattern altitude when you intercept the 45-degree track. If you expect to need more time to descend plan for a longer entry.


Enter the traffic pattern on a 45-degree-to-downwind.

Note that this is subtly different from your &quot;Option B&quot; (which has the descending turn leading directly into the downwind) in order to comply with the advisory circular's instruction that you should be &quot;well clear of the pattern before descending&quot;.
I cribbed the pattern entry graphic from Alliance Airways, with some minor modifications, as it was the best illustration of the technique I could find.