Definition of uniqueness of tubular neighbourhoods

Question

The tubular neighbourhood theorem states that if $M \subset N$ is an embedding of smooth manifolds without boundary and $\nu: E \to M$ is the normal bundle of $M$ in $N$, then there is a smooth embedding of the total space of $E$ onto a neighbourhood $U$ of $M$ in $N$ that restricts to the identity on $M$ (identified with the zero section in $E$). There is also a uniqueness claim: any two such embeddings of $E$ into $N$ are smoothly isotopic. I have difficulty with the latter claim. Suppose $M = S^1$ is embedded in the standard way into $N = \mathbb{R}^2$. The normal bundle is $\nu = S^1 \times \mathbb{R}$ and a tubular neighbourhood can be taken to be an annulus $U$ containing $S^1$; an explicit embedding is given by $f: (x, t) \mapsto x(1+\epsilon\arctan t)$ for any $\epsilon > 0$ small enough. Suppose $r: \nu \to \nu$ given by $(x, t) \mapsto (x, -t)$ is the reflection of the normal bundle. Then the claim is that there is an isotopy $F: f\simeq f \circ r$, which is clearly impossible. Could someone point out the mistake in my argument? Or does the uniqueness claim in the tubular neighbourhood theorem only hold up to linear automorphisms of the normal bundle?