I don't think there is a special Japanese pronoun in order to distinguish your second case. I don't think there is a special English pronoun, either.
私たちは, 私どもは, 我々は or "we" basically includes the two possibilities you mentioned.
There isn't such a pronoun, but we can distinguish them by some other techniques.
In a scientific paper, when the authors are plural, a strict reviewer/editor may dislike the pronoun, "we." They would proofread it to "our team" or something, because they want to exclude the listeners or readers from "we" and make it clearer.
Likewise, in order to exclude the listeners/readers, we can change them, for example:
私たちは ---> 弊社では (our company, in case that the listeners are the customers)
私たちは ---> 我々従業員一同は (all of our staff members, in case that the listeners are customers)
我々は ---> 私どもスタッフは (our staff members)
私たちは ---> 私たち日本人は (in case that the listeners are not Japanese people).
If you want to include the listeners,
私たちは ---> 私たちは皆様と共に
私たち ---> 私たち日本人は (in case that the listeners are all Japanese people)
我々 ---> 我々国民は in case that the listeners are citizens).
We can avoid the confusion by using specific expressions, but answering to your question, there is no such special pronoun. No.