the screen always shows it correctly. but you lose ownership of incomplete track if you don't continue extending it in your next turn (at least for one tile).
quotes from the rulebook:
[page 6] "You mark your ownership of a link with one of your tokens anywhere along the link when you start building a link."
"Track used to make a link can be built in one turn or over several turns."
[page 10] "If any track fails to reach either a town or city at the end of your turn, it is an “incomplete link.” An incomplete link consists of one or more track tiles that do not connect to a town or city at both ends. An incomplete link is still marked with your token, and other players may not extend it while you own it. However, you lose ownership of the incomplete link if you do not extend it on your very next turn. If you do not extend the track in the next turn (and every subsequent turn, until the link is completed), you must remove your token from the track. It then becomes an unowned, incomplete link."
any player can extend an unowned, incomplete link and will own it by doing so.
these rules for building track are the same in basic and standard games.