Hi Ted,
It basically depends on which app you use to receive the notifications. Here is a screen shot showing two notifications from different apps:
The one on top is from PHY, a third-part browsing app for the Physical Web. It doesn’t show any icon nor the URL of the page, but instead displays the content of the meta description tag.
The second notification is from Android’s built-in Nearby feature, which also scans for Physical Web beacons. It shows the icon, page title and URL, but omits the meta description text.
If you install the official Physical Web app, you will get a slightly different notification format:
(screen shot taken from the Google Play Store listing)
It could also be that your phone is using Chrome’s own implementation of the Physical Web to display the notifications, and for that reason you always see the Physical Web icon next to every notification.