It's so strange how the buttons work differently for all of us even if they look the same. When I click on the "Reply" button it opens up a "reply to thread" window with a large text box where I type my post, add smilies, etc. The text of the person whose post I am replying to is quoted and I can edit it. After I type my post in the text box, I click on "Submit Reply" and my post is appended to the end of the thread, where everyone can read it. It does not go as a private message to the person's mailbox. I do not have a button that does that. For me, the way to send a PM is to click on their name and choose from the dropdown menu that appears.

There is no need for a quick reply button for me because on my display, there is always a quick reply text box open at the bottom of the page, and you just start typing. No need to click a button. (I don't use this, I prefer the regular reply.)