Password protecting wp-admin problem – extra Login appears when just browsing

I password protected wp-admin in cPanel for added security. However, testing this while logged out (with various browsers) the result is that a very off-putting Login Box appears when anyone is just browsing my blog.

For a while I added this text to the box:
“You do not need to sign in to read this blog. Please, just click CANCEL to make this annoying box disappear”

Is there a way of adding extra security for wp-admin via cPanel that will not automatically pop-up this extra log-in box?

What I thought would happen, if it is possible and anyone can tell me how to do it, is for the extra log-in to appear only when the normal “Log in” link is selected. with entry of correct details then allowing access to the normal Log In screen

Best wishes,
CrlyWrly