Equivalence of Definitions of Tree

Proof
Let $G$ be a simple graph.

Definition 1 implies Definition 2
Let $G$ have no circuits.

As by definition a cycle is a circuit, it follows that $G$ has no cycles.

Definition 2 implies Definition 1
Let $G$ have no cycles.

$G$ contains a circuit $C$.

From Circuit contains Cycle as Subgraph it follows that $G$ contains a cycle.

From that contradiction it follows that $G$ can have no circuits.