You had me at the first paragraph of your reconciliation and lost me in the second. Let me try to be clear.
When we say we are 95% confident that the parameter is in the interval we are (as you said) not saying that the probability that the parameter is in the interval is 95%, but that the probability that this procedure yields an interval containing the parameter is 95%.
The distribution that this probability is coming from is the sampling distribution. 95% of samples will lead to an interval containing the (completely fixed) parameter.
edit
Everything about the parameter is deterministic. However, when you relate it to a random variable you can make probabilistic statements that are based on the distribution of that random variable.
The sample you draw is a random variable. Therefore the confidence interval you get is a random variable. So the answer to the question "Will the parameter be in the interval?" is a random variable. But the answer to "is the parameter in this interval?" is not.
To make an analogy. If I produce a coin that is heads 95% and tails 5%. And tell you it is 95-5 but I don't tell you which side is more likely. The following statement is true "If I flip the coin the probability it will land on the face it is biased toward is 95%." Once you flip the coin, the following is false "There is a 95% chance it is biased toward this face." You could declare "I say, with 95% confidence that the coin is biased to this face" because the word confidence in this field means precisely that!