A further approach would be to use an event of system type CONNECT for logging purposes. Note that events run on their own connection so they cannot "impersonate" the newly connected user but might be helpful to log the total number of current connections or some characteristics of the connection like the User or AppInfo information - available via the EVENT_PARAMETER system function.
You could use the corresponding "DISCONNECT" event to un-list the accoding user. (Or you could use the latter in conjunction with the suggested login procedure to update the row in your "current connections" table to mark the connection as closed, even if a connection is closed irregular.)