You've got the logic backwards; statement 2 implies statement 1. I think you're confused by the difference between statements being true, and statements implying each other.
That is, assuming that "continuous symmetry of the Lagrangian" is defined as "a vector $\vec K$ such that the substitution $\vec q \mapsto \vec q + \epsilon~\vec K$ for arbitrarily small $\epsilon$ leaves $L(q,\dot q, t)$ invariant," the second statement is an explicit construction of an object which is asserted to exist in the first statement.
Explicit constructions are more specific than general statements of existence. The first statement does not specify how the construction comes about, therefore if we were doing something that was not quite Lagrangian mechanics but had significant overlap, you might still have a Lagrangian in your system and might even be able to prove that its symmetries are one-to-one with conserved quantities: but possibly, that construction would not generate them.
This circumstance of explicit constructions being more specific than general statements of existence reaches a very fine point in some of the discussions around the axiom of choice in pure mathematics. For example the axiom of choice implies that there exists a well-order of the real numbers, in other words, a way to sort the real numbers so that every subset contains a least element. (The obvious ordering is not a well ordering because of intervals like $(0, 1)$ which do not contain their endpoints and intervals like $(-\infty, 0)$ which go off to negative infinity.) Nobody knows what such a well-ordering would look like; there is no plausible construction of the well-order, and the best we can do is to prove that the existence of the well-order is not necessarily unconstructable... it gets to be a bit of a mess. In addition to the mathematicians who don't like the axiom of choice because of this, there are some even-more-reactionary mathematicians who in addition don't like the idea of proof by contradiction for similar reasons; to their mind you should not be able to reason from "not-(not P)" to "P". This is not as bad as it sounds because there are some places where things like logic look very familiar, for example in type theory, where there is no easy equivalent to the "not-" predicate and those that do exist probably cannot admit a function of type
forall x. Negate (Negate x) -> x.