] I had always been of the understanding that the title "Viscount" was held by the heir of a landowning Lord who, once the Lord was deceased, then attained the title of Lord ____, and subsequently the new Lord's heir then became the Viscount.

Like so many things English, this is a trick they play to confuse outsiders. Here's a link to a basic introduction to the system of British titles. There are five ranks of nobility, from the top Duke, Marquess, Earl, Viscount and Baron. A man could hold any of those titles in his own right, which allowed him to sit in the House of Lords.

Your particular situation is this. As a family rose in rank through the aristocracy, the bearer of the title kept his the junior titles that he or his ancestors had picked up. For example, the Marquess of Anglesey was also the Earl of Uxbridge and Baron Paget.

It was customary for the oldest son of a peer to use one of the junior titles as his courtesy title, as long as it was different from his father's. For example, the Duke of Ditchwater might also be Viscount Gutter and Baron Puddle. If so, his eldest son would be called Viscount Gutter until his father died and he inherited the Dukedom. The courtesy title didn't allow him to sit in the House of Lords.

Here's an illustration. British Admiral Horatio Nelson was created Viscount Nelson. That was his own title. British General the Duke of Wellington had, a few years before he was made a duke, been given the lesser title of Viscount Douro. When he was a Duke, his eldest son was called Viscount Douro as a courtesy title.