Since version 13.2.49.78 the rows look like those of Conversations and stay compact: on the left the site's icon or its initials, with no box around it. Since version 13.2.49.77 the row has a single line (that version also made it half as tall): the icon, the project's name (the web address) and, right after the name, a light grey number with how many updates you have not read yet in the project's Planning, Rules, Conversations and Activity (99+ above 99, nothing at 0); there is no line under the name any more. Inside the project the same total is split across the tabs: the grey numbers next to Planning, Rules, Conversations and Activity add up to the number on the row. In a project where you are the only person Conversations is not there and does not count, and a project you have never opened starts counting as soon as SyncNow follows it for its notifications, within a minute or so of signing in, and what was there before counts as read. The number stays even when Show notifications is unticked for the project: that choice only takes it out of the badge on SyncNow's taskbar button and on its icon in the notification area, and silences its notifications. Click anywhere on the row to open the project; with the keyboard, move to the row with Tab and press Enter.
A badge appears only for something out of the ordinary: Access revoked when your access was removed (by the owner, or because the project was removed); Not verified when the server has not confirmed the project's data recently. A yellow sign in place of the icon means a connection problem, and clicking it explains the reason. A small green dot next to the name means someone else is working on the project right now; pointing at it says Someone is working on this project. No other mark on the row shows who is online or offline, and a name too long for the row ends with ... before the number, the full name showing when you point at it.
An arrow on the right says the row opens. When you point at the row, or reach it with the keyboard, two small buttons appear: Activity, which opens the project's Activity, and ..., with Activity, Connections, Project settings and Remove project (Leave project on a project you were invited to). Under the pointer they only turn green: no tooltip and no coloured square behind them. A right-click, or the Menu key, opens the same menu. Where the line that used to be under the name went is explained in Where did the line under a project's name go?.