Briefly: According to the paper you are linking, Abl-BCR is not "recognized" by the drugs in the active or the inactive state but rather it assumes a conformation that is similar to the active or inactive state of Abl once the drug binds. Simply put: the paper does not study what conformation Abl was in before the drugs bind, they rather discuss the conformation after the binding of the drugs.
Just to provide some background: Abl is activated by fusion to BCR because the regulatory part of Abl gets damaged such that the tyrosine kinase is constituously active. This damage is in the regulatory part of Abl and not in the nucleotide binding pocket (the part where the drugs bind and the part that strongly changes conformation between the active and inactive states).
Both imatinib and dasatinib (like other kinase inhibitors) are competitive inhibitors that bind to the ATP binding pocket. This means that both drugs occupy the ATP binding site, preventing ATP from binding and the ABL protein from executing its kinase activity no matter whether it is in the active or inactive state.
The difference is that imatinib binds to (and blocks) the pocket where normally the phosphate groups of ATP bind, forcing Abl (and Abl-BCR) into a conformation that is similar to the inactive state. The drug can still bind to the active state but upon binding, Abl changes its conformation and then looks similar to the inactive state.
Dasatinib on the other hand binds to (and blocks) the part where normally the base of ATP binds. 171
This part does not strongly change its conformation between active and inactive state and thus dasatinib should be able to bind to both active and inactive conformation. However note that the ability of dasatinib to bind to either conformation is simply a hypothesis they state in the paper and is not substantiated by data (in fact they themselves say that the data indicates that dasatinib binds stronger to the active conformation).
Note no matter which conformation the protein has (active or inactive) it cannot perform its function as soon as any of the competitive inhibitors binds because the inhibitors block the ATP binding site and without ATP, Abl cannot phosphorylate anything because ATP is the source of the phosphate group which Abl would normally transfer to its target.
Some more background on how Abl works: Abl is a tyrosine kinase which means that it transfers a phosphate group from ATP to a tyrosine residue to its target protein. Also: Abl has some regulatory parts that control whether it performs its tyrosine kinase activity. In the inactive state it will not do anything even in the presence of ATP and its target protein. In the active state, it still needs ATP and its target protein to perform its kinase activity. 171