It all depends on whether you are looking at the probability conditional or unconditional on the data. Suppose you have an unknown parameter $\theta \in \Theta$ and you make a confidence interval for this parameter using sample data $\mathbf{x}$. Let $\text{CI}_\theta(\mathbf{X},1-\alpha)$ denote the (random) confidence interval at confidence level $1-\alpha$ and with (random) data $\mathbf{X}$. An exact confidence interval satisfies the following conditional probability condition:

$$\mathbb{P}(\theta \in \text{CI}_\theta(\mathbf{X},1-\alpha) | \theta) = 1-\alpha \quad \quad \quad \quad \quad \text{for all } \theta \in \Theta.$$

If we are willing to ascribe a probability distribution to $\theta$ (e.g., as in Bayesian analysis) this also implies the marginal probability that:

$$\mathbb{P}(\theta \in \text{CI}_\theta(\mathbf{X},1-\alpha)).$$

However, it is not true that:

$$\mathbb{P}(\theta \in \text{CI}_\theta(\mathbf{X},1-\alpha) | \mathbf{X} = \mathbf{x}).$$

As you can see from the above, if we are looking at the probability unconditional on the data (and either conditional or unconditional on the parameter) then we can say that the probability of the unknown quantity falling into the confidence interval is equal to the confidence level. However, if we are looking at the probability conditional on the data we cannot say that the probability of the unknown quantity falling into the confidence interval is equal to the confidence level. Typically, we frame this by saying that the confidence interval procedure/method (considered prior to substitution of the data) will cover the true parameter with probability equal to the confidence level, but once we have an actual confidence interval (i.e., after substituting the observed data and conditioning our probability statements on the data) this probability statement no longer holds. This is the reason we refer to having 95% "confidence" rather than 95% probability for the parameter being in the interval.

It all depends on whether you are looking at the probability conditional or unconditional on the data. Suppose you have an unknown parameter $\theta \in \Theta$ and you make a confidence interval for this parameter using sample data $\mathbf{x}$. Let $\text{CI}_\theta(\mathbf{X},1-\alpha)$ denote the (random) confidence interval at confidence level $1-\alpha$ and with (random) data $\mathbf{X}$. An exact confidence interval satisfies the following conditional probability condition:

$$\mathbb{P}(\theta \in \text{CI}_\theta(\mathbf{X},1-\alpha) | \theta) = 1-\alpha \quad \quad \quad \quad \quad \text{for all } \theta \in \Theta.$$

If we are willing to ascribe a probability distribution to $\theta$ (e.g., as in Bayesian analysis) this also implies the marginal probability that:

$$\mathbb{P}(\theta \in \text{CI}_\theta(\mathbf{X},1-\alpha)) = 1-\alpha.$$

However, it is not generally true that:

$$\mathbb{P}(\theta \in \text{CI}_\theta(\mathbf{X},1-\alpha) | \mathbf{X} = \mathbf{x}) = 1-\alpha.$$

As you can see from the above, if we are looking at the probability unconditional on the data (and either conditional or unconditional on the parameter) then we can say that the probability of the unknown quantity falling into the confidence interval is equal to the confidence level. However, if we are looking at the probability conditional on the data we cannot say that the probability of the unknown quantity falling into the confidence interval is equal to the confidence level.

Typically, we frame this by saying that the confidence interval procedure/method (considered prior to substitution of the data) will cover the true parameter with probability equal to the confidence level, but once we have an actual confidence interval (i.e., after substituting the observed data and conditioning our probability statements on the data) this probability statement no longer holds. This is the reason we refer to having 95% "confidence" rather than 95% probability for the parameter being in the interval.