Imagine I have a test for a disease that returns either a positive or a negative result. I administer the test to one group twice in a single session; in another group, I administer it twice with two weeks separating the tests. The hypothesis is that the tests taken at the same time will be more consistent with each other than the tests taken two weeks apart. How can I test this?
Bonus question: What if the two groups consisted of the same people?
Both situations are specific cases of test-retest, except that the recall period is null in the first case you described. I would also expect a larger agreement in the former case, but that may be confounded with a learning or memory effect. A chance-corrected measure of agreement, like Cohen's kappa, can be used with binary variables, and bootstraped confidence intervals might be compared in the two situations (this is better than using $\kappa$ sampling variance directly). This should give an indication of the reliability of your measures, or in this case diagnostic agreement, at the two occasions. A McNemar test which tests for marginal homogeneity in matched pairs can also be used.
An approach based on the intraclass correlation is still valid and, provided your prevalence is not extreme, should be closed to
About your bonus question, you generally need 3 time points to separate the lack of (temporal) stability -- which can occur if the latent class or trait your are measuring evolve over time -- from the lack of reliability (see for an illustration the model proposed by Wiley and Wiley, 1970, American Sociological Review 35).
Perhaps, computing the tetrachoric correlation would be useful. See this url: Introduction to the Tetrachoric and Polychoric Correlation Coefficients