Does "the weed" refer to tobacco or marijuana?

Solution 1:

As is noted in the question, the Longman Dictionary of Contemporary English defines the weed as:

informal cigarettes or tobacco

However, the same dictionary defines weed (without the article) as:

6. [uncountable] cannabis

So it would appear that there's a clear distinction between the count and non-count usages.

The question goes on to ask:

But "weed" nowadays is also closely associated with marijuana. If I am using "the weed" in my own writing, would native speakers take it to mean "tobacco"?

It will depend both on context and whether it's being used correctly. Most people are likely to distinguish between

I'm trying to give up the weed

and

I'm trying to give up weed.

But a sentence like "have you got the weed?" could be ambiguous without a suitable context.

Solution 2:

There is an article here which discusses when the term weed began to be used as a slang term for marijuana and concludes that it was in the 1930s. Although it further claims that this name didn't become popular until the 1990s.
The word marijuana entered the English language in the late 19th century both according to the above article and Wikipedia.
On the other hand the name cannabis dates back to the 16th century according to Wikipedia.
I am prepared to accept that weed as applied to tobacco dates from the early 17 century.
Just how and why the old name for tobacco got applied to marijuana is unclear.