Hi Jim,
Since reading your email, I've been playing around with line lyrics. I've never used them before and I appreciate your inspiring me to investigate them.
When I started reading about them, I noticed that the "/" you refer to is used for note-based entry and not line-based entry.
I have BIAB 2014, and how I entered the line lyrics is as follows.
1. I went into Notation Editor. There I clicked on the Notation Editor options (#1) and set the "Bars/Screen" to 4 (#2). This will cause the on-screen display to have four bars. Then I exited.

2. Next, I typed my lyrics in the text line at the top of the notation window.

3. When I had typed the lyrics for the 4 bars that were showing on-screen, I hit the DOWN arrow on the computer keyboard. This moved me to the next block of four bars.
I repeated this process of typing four bars worth of lyrics and hitting the down arrow until I'd finished the song.
4. After the lyrics were entered, I then selected "Lyrics" on the top menu (#1) and chose "Copy lyrics to clipboard" (#2) from the dropdown menu.

After that, I went back to the "Lyrics" menu and this time selected "Lyric Document Window" (#3).
5. Once the lyric document window had opened, I clicked on the "Paste from clipboard" icon.

6. This gave me the following ...

Is this the sort of thing that you are after? (Note: the top four bars are blank because that's the introduction.)
Regards,
Noel