Mike! You're right! I had never noticed the little red thingies!
The reason, I think, is that I had been looking in the wrong place (or the place I'd been looking wasn't telling me what I wanted). All this time I've been going to the forum Home screen, expecting to get an overview of recent activity. But that level--I see now--does not show your little red thingies.
But you've shown me to click TWO LEVELS FURTHER IN, to where they DO appear, just as your screenshot shows.
ta-DA!
Also, if I let the pointer hover over a red thing it shows "Go to first unread post," but if I hover over the link to its right I see a quotation from that post.
And the link itself (visible in Firefox at the lower left of the display) is different, depending on whether the pointer is hovering over the red thing or the link.
I had missed all that. The only disadvantage is that you have to select either each category of discussion first. So all I wish for now is that the red things be available in the "Home" screen.
The browser's Back button doesn't update the red things. You have to reload the page. But that's easy enough because the path is displayed in the upper left, so you can just click on the second-to-right item to go up one level. It could be easier, but it does work.