HBr and HCl are strong acids, but HCN is actually a fairly weak acid. I did not look at your link, so I cannot comment.
When it comes to the three definitions for acids and bases, the reality is that acids are notably different but bases are very similar in their behavior. The crux of the difference between the various definitions lies with the solvent.
Arrhenius is used when the solvent is water. Any compound that donates an H+ to water to form H3O+ is acidic and any compound that takes an H+ from water to form OH- is basic. While Br- and Cl- are close to being inert in water, we can still think of them as conjugate bases of their respective acids, so they will make a very small (essentially immeasurable) amount of hydroxide. CN- will make a detectable amount of OH- in water, so it is in fact an Arrhenius base.
The Brønsted-Lowry definition is applicable in any polar solvent, including water, where an acid donates H+ and a base accepts H+. The three aforementioned anions will accept H+ to some degree, making them Brønsted-Lowry bases.
The Lewis definition is applicable in any solvent, and is commonly applied to organic solvents. An acid is a lone pair acceptor and a base is a lone pair donor. A nonpolar species such as BF3 does not agin or lose an H+ , but it can mimic the behavior of H+ in a nonpolar environment, accepting an electron pair (like H+ does in protic solvents).
So to answer the original question of what kind of base Br-, Cl-, and CN- are, the answer is "any kind you want them to be, depending of the solvent you use them in."
