Neither quite work as a list of assumptions for the t-test.
- The population is assumed to be normally distributed
- Samples are random
okay so far
- If there is deviation from either of the above, sample size must be larger.
This is NOT an assumption of the t-test. It's advice about what you need if the assumption of normality isn't met (it's of no use if assumption 2 doesn't hold though), and then it's not $t$ in any case; you'd have to invoke two results to argue the statistic would be asymptotically normal.
- $X$ follows a normal distribution with mean $\mu$ and variance $\sigma^2$
Well you actually also need independence. Once you add that (which is related to your "random sampling" assumption), you're set.
- $s^2$ follows a $χ^2$ distribution with $p$ degrees of freedom under the null hypothesis, where $p$ is a positive constant
This is a consequence of the first assumption (once you add the missing independence)
- $Z$ and $s$ are independent.
This is also a consequence of the first assumption (again, once you add the missing independence)