it is no different than usual. However molecular methods seem better. Kindly share regarding pulmonary or specific extrapulmonary TB you are interested in.
To answer your first question, the techniques won't be different regardless of the patient's HIV status. But depending on what is available, I would recommend you use TB liquid culture (MGIT) or solid culture (LJ or M7H11 agar plates). The GeneXpert MTB/RIF as well as the ZN stain can be used but culture remains the gold-standard for TB diagnosis.
With regards to " But is Mycobacterium tuberculosis that cause the tuberculosis in HIV patients or different species? "
During my time researching into TB. The answer is not really, however HIV+ patients are susceptible to Non-tuberculosis Mycobacteria NTM, but keep in mind that NTM are considered opportunistic infections.
Now there are other strains that can cause TB called Mycobacterium tuberculosis complex, however this relies whether the patient was exposed to the host organism, did the patient eat under-cooked meat or drink unpasteurized milk.
My recommendation would be to culture the organism and perform spoligotyping to identify the strain.
1) For the first question, the diagnosis of Pulmonary tuberculosis in HIV + patients depends on the availability of tests in your area and also on the degree of immuno-depression. Knowing that smear microscopy is less sensitive than Xpert and Culture( find attached document1). Also severe immuno-depressed individuals would more likely to be smear negative- Xpert and/or Culture positive (now you will need a clinical approach to put on treatment early enough).
2) Among the HIV+ individuals, tuberculosis seems to be the the first opportunistic infection be it MTB or NTM depending on the immunological state of the patient and the living environment.
3) The diagnosis of NTM is been made using Culture where it is possible to do the identification of the colonies using different techniques (find attached a document on NTM) like Maxwell previously said.