I think in this section, you only need to use the name field to reference the row and not your alternateurl.
// Store a reference to the table row so we can get it again later.
htmlForPath[scoreSnapshot.name()] = newScoreRow;
htmlForPath[scoreSnapshot.AlternateURL()] = newScoreRow;
get rid of the last line.
Unless there is an error somewhere else, you've addressed the areas to change/add code (variable names are my version:
<input type="text" id="urlInput" placeholder="URL">
newScoreRow.append($("<td/>").text(scoreSnapshot.val().url));
var url1 = $("#urlInput").val();
userScoreRef.setWithPriority({ name:name, score:newScore, url:url1}, newScore);