Firstly, it is not entirely correct to say that you cannot make a probability statement about a confidence interval. The set function that defines a confidence interval is actually defined by an a priori probability requirement, expressed as a probability statement that conditions on the unknown parameter but not the data. Viewed as a procedure (i.e., as a function operating on data and a chosen confidence level) it is possible to make a valuable probability statement about the confidence interval, as we will see below (see here for more formal discussion of the definition).
Suppose we have an observable data vector $\mathbf{X} \equiv (X_1,...,X_n)$ from a distribution with some unknown parameter $\theta \in \Theta$. Then a confidence interval procedure for the parameter $\theta$ is a set function $\text{CI}$ that satisfies the following conditional probability requirement:$^\dagger$
$$1-\alpha = \mathbb{P}(\theta \in \text{CI}(\mathbf{X}, \alpha)|\theta)
\quad \quad \quad \text{for all } \theta \in \Theta \text{ and } 0 \leqslant \alpha \leqslant 1.$$
Note that for any prior distribution over $\theta$, this also implies the weaker probability result:
$$1-\alpha = \mathbb{P}(\theta \in \text{CI}(\mathbf{X}, \alpha))
\quad \quad \quad \text{for all } 0 \leqslant \alpha \leqslant 1.
\quad \quad \quad \quad $$
The probability requirement above is an a priori probability statement that treats the data (and therefore the confidence interval) as random. Now, given a set function that comports to the above conditional probability requirement, for a given value $0 \leqslant \alpha \leqslant 1$, the confidence set for the observed data $\mathbf{x}$ (with confidence level $1-\alpha$) is the fixed set $\text{CI}(\mathbf{x}, \alpha)$. This object is no longer random, so it now either contains the parameter $\theta$ or it does not.
Properly interpreting a confidence interval: The proper interpretation of the confidence interval hinges on correctly stating the conditioning information in the probability statement. As can be seen, the probability requirement for a confidence interval is conditional on the parameter, but it is not conditional on the data (i.e., it is viewed a priori, where the data are random variables yet to be observed).
Confidence intervals can be properly interpreted either by correctly stating their probability requirement, or by stating the analogous long-run behaviour of the interval for hypothetically repeated data using the "law of large numbers". Here are some examples of correct and incorrect interpretations of a 95% confidence interval:
(Correct): When viewed as a procedure that has not yet been applied to the data (i.e., not conditioning on observed data), there is a 95% chance that the parameter falls within the confidence interval.
(Correct): Regardless of the true parameter value, the prior probability that the confidence interval procedure will lead to a confidence interval containing the parameter is 95%.
(Correct): If the confidence interval procedure is applied repeatedly to random data from the stipulated distribution (the one used to form the confidence interval), in the long-run the parameter will be in the confidence interval 95% of the time.
(Incorrect): The confidence interval formed for a given set of observed data will contain the parameter with 95% probability.
The problem with the statements given by your teacher is that they are ambiguous about what probability statement is being made (and they both seem to say the same thing in a different way). When viewed as a fixed interval after the data are observed, it is true that the interval either contains the data or does not contain it (with probability zero or one). However, when viewed as a procedure without conditioning on observed data,
I would recommend you try to get into the habit of distinguishing an actual (fixed) confidence interval formed from observed data from the confidence interval procedure that maps the data to the interval. The latter is the object for which a valuable a priori probability statement can be made.
