I wonder whether the T-spot blood test is sensitive enough to determine difference between exposure to the tuberculosis and tuberculosis vaccination.
The T-spot blood test for tuberculosis is an interferon-gamma release assay (IGRA or IFN-γ assay). There are two of these tests (Quantiferon and T-spot) with small differences, but the basic mechanism is the same and they are treated the same for the CDC, which writes:
IGRAs, unlike the TB skin tests, are not affected by prior BCG vaccination and are not expected to give a false-positive result in people who have received BCG
BCG is the tuberculosis vaccine.
While I couldn't find references to primary literature on that CDC site, what I found myself supports this
The paper Discrepancy Between the Tuberculin Skin Test and the Whole-Blood Interferon γ Assay for the Diagnosis of Latent Tuberculosis Infection in an Intermediate Tuberculosis-Burden Country concludes
the IFN-γ assay based on the ESAT-6 and CFP-10 M tuberculosis–specific antigens is a useful method for detecting latent TB infections and might help to eliminate the limitations of the TST [tuberculin skin test] in BCG-vaccinated populations in intermediate TB-burden countries.
Tuberculosis contact investigation with a new, specific blood test in a low-incidence population containing a high proportion of BCG-vaccinated persons reaches the same conclusion and recommends the test for vaccinated people. That study used the Quantiferon test, but the two tests have roughly the same sensitivity.