"Direct Message" and "Private Message" indeed mean different things. In practice, because both involve messaging one individual user, a good deal of people (including myself) still expect them to be functionally the same. Part of this functionality we expect is that there is an attempt to make these messages less visible and easy to access than the reply I just sent to you right now. This expectation is validated on Twitter:
Direct Messages are the private side of Twitter. You can use Direct Messages to have private conversations with people about Tweets and other content.
a private message sent on a social media website, that only the person it is sent to can see
and by the fact that if you go on anyone's profile, you can see post history, comment history, and boosts, but not a list of who they tried to send an individual message to or what those messages were. I believe that more technical people could retrieve such messages, that the messages are not totally secure, but to my layman eyes, I do still expect that there was at least an attempt to make these messages private.