The lawyer is a second party, not a third party. In the telephone call situation, you are calling someone else, using the phone company's (the third party's) equipment.

In any case, there may be attorney–client privilege, but it only applies if the lawyer is acting primarily as an attorney for you. If you are using the lawyer just to store documents that you don't want to keep yourself, then there's no privilege.

So what does that make cloud service providers like dropbox? Aren't they directly analogous to me giving a document to someone to put in a safe?

I don't understand the question, sorry. Dropbox isn't a third party, and it isn't your attorney (or a handful of other fields with special legal status).

You should look to see the legal precedent for private courier service, or for storage facilities. Those are more likely to be comparable to dropbox, if only because consumer protection laws may be relevant.

(That is, you can't give it to your friend, to put into a safe, and expect that it has any status different that anything else that your friend had. For example, a warrant may require your friend to open the safe, because your friend might also store personal items along with your items. While for a storage facility, a warrant to search the business's records would not automatically imply searching the entire building.)