Fortunately there is an Interacoustics Academy webinar which is aimed at addressing this very topic, and covers in detail several of the tests you mention above – i.e. their principle and clinical applications. Please see “Advanced Tests in Audiometry Parts 1 & 2”.
The short answer is that there are a number of test options included in many models of audiometer (clinical and diagnostic) for use in specific circumstances where routine Pure Tone Audiometry and/or Speech Audiometry is not sufficient to reach a diagnosis or guide audiological management decision making.
Stenger = a test with binaural sound presentation, which utilises a phenomenon of binaural interaction to indicate from which ear a sound is being perceived. This test is useful in cases of suspected unilateral non-organic hearing loss.
Langenbeck = This is a suprathreshold test where noise and pure tones are presented to the same ear, and the intensity of the pure tone that can be discriminated from the noise is measured in order to assess the masking pattern needed to mask a sound.
Weber = A test of sound lateralisation when test signals are presented by bone conduction. This test is useful in identifying asymmetrical hearing loss as the sound will tend to lateralise to the better hearing ear.
Alternate Binaural Loudness Balance (ABLB) = a test with sounds (typically pure tones of the same frequency) alternated between ears. Those in one ear are adjusted in level to match the other for loudness – at which point sounds are said to be balanced. Amongst other applications, this test is useful as part of a test battery for diagnosing retrocochlear disorders. In particular, it is used as a preliminary test for Auditory Brainstem Response interaural latency and intensity comparisons.
Short Increment Sensitivity Index (SISI) = This tests the ability of the listener to detect small increases in loudess. This helps identify recruitment and, amongst other applications, this test is useful as part of a test battery for differentiating cochlear from retrocochlear disorders.