Do you capitalize Lake when you are referring back to proper noun Tahoe Lake [duplicate]

From the web site tahoe.com:

Lake Tahoe, the clear, cobalt blue lake tucked into the snowy Sierra Nevada mountain range, welcomes athletes, adventurers and casual travelers to its easygoing and pine-scented atmosphere. A long sought-after vacation destination, Lake Tahoe keeps everyone’s interests piqued and their vacations active throughout the year. Lake Tahoe is the nation’s second deepest lake, and the area previously served as a transcontinental route for first the railroad and later highways.

From the web site Lake Tahoe Travel Guide:

... While the stunning blue lake alone is worth a trip, the surrounding area, also known as Lake Tahoe, boasts miles of hiking trails, dozens of picture-perfect vistas and some of the best skiing in North America.

It should be "lake" with small l. When "lake" refers back to "Lake Tahoe", it is not referring to part of the name "Lake Tahoe", but rather to the referent of "Lake Tahoe", which is a lake.

Suppose that Lake Tahoe was actually the name of a small pond -- then the name "Lake Tahoe" wouldn't be totally appropriate, but maybe it was originally named when it was big, then for some reason, most of it dried up. Then you wouldn't think of referring to it as a lake, because it isn't one. You would refer to it as a pond.

So the capitalization of a definite reference depends on what the thing is, not what it's called.