The statement $$(x<3)\to (x\ge 3)$$ is not a contradiction, i.e. it is not ALWAYS false. It is merely false for $x<3$. If $x\ge 3$, then it is vacuously true. It contains a free variable, so we can't say that it is false.

The implications are confusing the issue. Consider the statement $x<3$ alone. It is not necessarily true. Also, the statement $x\ge 3$ is not necessarily true. But their disjunction is true.

The statement $$(x<3)\to (x\ge 3)$$ is not a contradiction, i.e. it is not ALWAYS false. It is merely false for $x<3$. If $x\ge 3$, then it is vacuously true. It contains a free variable, so we can't say that it is false.

The implications are confusing the issue. Consider the statement $x<3$ alone. It is not necessarily true. Also, the statement $x\ge 3$ is not necessarily true. But their disjunction is true.

The OP's edit makes no difference. What appears to be confusing OP (and confuses many learners of mathematics) is that $p\to q$ is true when $p$ is false. What I find helps is the theorem that $p\to q$ is equivalent to $q\vee \neg p$.