Actually, any unsupervised method, including clustering, still has training data. The only difference is it doesn't have a target y variable in the training set

Actually, any unsupervised method, including clustering, still has training data. The only difference is it doesn't have a target y variable in the training set to minimize the error metric, hence the name unsupervised. But the definition you mention is right. Yet, any dataset that you use to fit your model will be your training set, even if you don't have a train test split or the like, because you used it to train your model over.