Neither quite work as a list of assumptions for the t-test.

1. The population is assumed to be normally distributed
2. Samples are random

okay so far

3. 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 (CLT + Slutsky) 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)

What makes something an assumption?

In order to work out the sampling distribution of the test statistic under the null hypothesis, some assumptions are required. In the case of the "standard" t-statistics (the ones that actually have t-distributions if the conditions hold), if you have independent, identically distributed normal observations (or iid normal pair-differences in the case of a paired test), then the t-statistic will have a t-distribution.

Neither of the lists you give quite work as a list of assumptions for the t-test.

1. The population is assumed to be normally distributed
2. Samples are random

okay so far

3. 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 (CLT + Slutsky) 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)

Neither quite work as a list of assumptions for the t-test.

1. The population is assumed to be normally distributed
2. Samples are random

okay so far

3. 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)

Neither quite work as a list of assumptions for the t-test.

1. The population is assumed to be normally distributed
2. Samples are random

okay so far

3. 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 (CLT + Slutsky) 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)