The language you refer to is the same in economics and in mathematics, for the simple reason that economics uses this language borrowed from mathematics, as it uses mathematics.
So, you have to look at the language of mathematics. The question of mathStackExchange you linked gives you important explanations.
I can just add something to what is said in an answer in mathStackExchange, a paragraph of which I quote:
A theorem is a logical consequence of the axioms. In Geometry, the "propositions" are all theorems: they are derived using the axioms and the valid rules. A "Corollary" is a theorem that is usually considered an "easy consequence" of another theorem. What is or is not a corollary is entirely subjective. Sometimes what an author thinks is a 'corollary' is deemed more important than the corresponding theorem. (The same goes for "Lemma"s, which are theorems that are considered auxiliary to proving some other, more important in the view of the author, theorem).
I want specify something about propositions and theorems.
It is true that propositions and theorems are conceptually the same thing: they are derived from the axioms through the inference rules, they are 'proved'.
But when you read a text of mathematics, some statements are called 'Theorem' and other 'Proposition'. This is a convention.
Usually, the name 'Theorem' is reserved to important theorems, often, but not necessarily, they have a name: 'The Hahn-Banach Theorem, the 'Theorem of contractions', and so on.
The name 'Proposition' denotes a less important theorem, which doesn't deserve the name 'Theorem'.
A Lemma usually is a theorem that is important to prove another theorem, and not very important on its own, but it is possible to find Lemmas considered important, for example in differential equations theory there is the 'Gronwall's Lemma', or 'Fatou's Lemma' in real analysis, which are considered very important theorems on its own.
A conjecture is just a hypothesis that a statement is true, but it hasn't been proved (or disproved): the Riemann conjecture, the Fermat conjecture (this last conjecture has been proved).