I think you are confusing track and line. "Track" refers to the physical attributes of the railway, so you have single-track, double-track, etc. which refers to the number of pairs of rails. The term "line" is generally associated with a method of working, e.g. "up line" refers to a track normally used for up trains, a "reversible line" is one that does not have a normal direction of working. "Single line", "Bi-Directional line" and "double line" likewise refer to different methods of working: with double-line working, one line is generally used for up trains and the other line for down trains.
The words Up, Down, Up & Down, Down & Up all mean that that is the direction of travel in an authorised signaled move. Up & Down and Down & Up refer to where the line starts and the normal direction of travel first.
Reversible and bi directional is not the same thing either.
A bi directional line is a line that is signaled in both directions of course but they are both fully controlled and fully signaled routes from peg to peg to peg.
Liverpool Central station is a rather good prototype to explain so grab your LNW Sectional Appendix. NW8001 004 Hunts Cross West Jn to Southport.
You will see the Up Southport (USP). it is bi directional from the Reversing Siding (RS) all the way out to Bootle Jn. (Notice how it is not called the Up & Down Southport)
The line has a also reversible but.
On the diagram, unfortunately it doesnt show pegs, but if it did you would see that on the Up Southport line just in advance of the switch for the reversing siding there is a position light signal to control a move from it back into either of the platforms (route indicator provided). The train drives onto the track and comes to a stand in rear on the signal (ML1017), it can then be given a reversing signal back into the platform.
At that point the track is not bi-directional, but it is reversible as there is a signal provided to perform a reversing movement that would otherwise be a bang road movement and thus not possible.
The at Liverpool South Parkway there is on the Up Fast 9809 signal.
The line is not bi directional, a train arrives in a move from 3860 - 3858. The next move is from 3858 to 3854 on the Up Fast in a forward direction but (There are two other moves in a forward direction possible too but I obfuscate them for ease of explanation) it is possible to reverse on that line too in a signaled move from 9809 to one of three possible signals, the most obvious of which is 3861 on the Down Fast.
That means that that line is reversible at that point.
Over on platform 3 the situation is the same, the line is reversible, one arrives in an Up direction but can reverse on 9807 signal to 5833 signal.
While on the Down Fast on platform 4 the line is bi directional. One stops in rear of 5831 signal and can leave the platform in the forward direction (opposite direction to that in which you arrived, but its still forward) and go over the right hand facing switch to Garston Junction and receptions sidings for the Freightliner terminal at Garston.
All single lines are bi directional, but not all bi directional lines are single lines and use "The working of a single line" rules. And then some lines have a single direction of travel but allow for reversals.
A bi Directional lines signals are all interlocked into each other, a signaled move in one direction would interlock with a conflicting move in the opposite direction to prevent a head on collision. But there is no cost cutting involved in its operation. A single line with electric token block allows a single line to be used in either direction with minimum staff levels... A good case in point is Buxton's line up to Hindlow. There is a box one end but its no signalman the other, as Buxton's operations require a signaler where as Hindlows yard doesnt.
A single line requires a method of controlling access to the single line that is interlocked to the method of controlling access at the other end of the single line but those methods of control do not have to be a signal box. It can simply be a Tyer's electric token machine in a shed. The release of a token requires somebody to give an assurance that the line is not in use and will not be used while the train is travelling along it and to assure this the tokens release is interlocked.
At Buxton there are two ways to access the single line to Hindlow, from the signal on the Up Main (Gives access to either the single line, the station or what used to be the diesel depot) or from the signal in the sidings which can also be accessed in two ways, either from the Up & Down Great Rocks or from the station/depot...
If the signaler wants to route you to Hindlow and once a token is withdrawn for the line, the points set and locked and signals pulled off the line, the physical stretch of track that lays between the signal post and the points that are the start of the track that is by itself without a neighboring road on either side of it becomes a single line and the "The working of a single line" rules apply to it. That is in-spite the fact that is is the Up Main.