You can't say "the Elvis Presley singer" to mean "the singer named Elvis Presley".
For rivers, the word "River" tends to come last. E.g. in the United States the "Mississippi River" and "Missouri River" are pretty much never named with the reverse order. But it depends on the specific river; the Thames is one where "the River Thames" is common.
"the Lisp programming language" and "the programming language Lisp" mean the same thing, and are both pretty normal word orders.
"the variable x" seems like a more usual word order to me. "the x variable" would probably show up most often in contexts where it is used to mean "the variable on the x-axis".
I think that your question covers two distinct constructions:
the attributive construction. In this construction, a noun, "N-bar" or "nominal" (sorry, I don't know a good, non-confusing term for this, but I wouldn't call it a "noun phrase" because it cannot contain an article) is used to modify a following noun, "N-bar" or "nominal"; the combination acts as an "N-bar" or "nominal" and can be preceded by an article. The meaning of an "attributive" construction is vague: it tells you that the head noun is of a "type" that is somehow related to the attributive noun, but it doesn't tell you the nature of the relationship. "Baby oil" is oil for using on babies, but "olive oil" is oil made from olives.
Apposition, a construction where two noun phrases that identify the same entity but in different ways are placed next to each other.
"The Star Wars movie" is an attributive construction: it brackets as "the [[Star Wars] movie]", and means something like "the movie of a type related to Star Wars". This doesn't necessarily mean "the movie named Star Wars": for example, during the time frame when the movie titled Rogue One: A Star Wars Story was playing in theaters, you might have heard someone say "I'm planning to go to see the Star Wars movie with my family tomorrow".
"The x variable" also seems to me to be most naturally interpreted as an attributive construction, bracketed as "the [[x] variable]"; this may explain the conjecture about its meaning that I mentioned above, where it would be expected to mean "the variable on the x-axis" = "the variable of a type related to x (= related to the x axis)".
I suppose that "the [[Lisp] programming language]" and "the [[Thames] River]" (or "river"?) are also attributive constructions, although I'm less sure about these.
"The movie Star Wars" is not an attributive-noun construction and it is not bracketed as "[the [movie [Star Wars]]]". Rather, is an appositional construction, bracketed as "[the movie] [Star Wars]".
I think that "[the singer] [Elvis Presley]", "[the River] [Thames]", "[the programming language] [Lisp]" and "[the variable] [x]" are also appositional constructions.
The proper-noun component does not always come first in an appositional construction:
John the Baptist (the whole thing here functions as a larger proper noun)
Elvis Presley, the singer, …
Lisp, the programming language, …
There seem to be some restrictions: neither "Thames, the river" nor "x, the variable" sound acceptable to me, and "Star Wars, the movie" sounds awkward. When the non-proper-noun part is longer, it may sound more acceptable to put it second: "x, the variable that we will be examining next," or "Star Wars, the movie that started a popular franchise" sound better. Likewise, something like "Elvis Presley, the famous singer, …" sounds more natural than just "Elvis Presley, the singer, …".