Because regular paths can be very wild indeed, it seems to me that Andreas' answer in the comments above is in fact the most intuitive definition...! (call two regular paths $p, q$ path-equivalent iff $p=q∘ϕ$ for some monotone increasing self-homeomorphism $\phi$ of $[0,1]$).

But from your (the OP's) comments I gather that you would prefer a definition that focuses more (also) on the path-images. Perhaps you're looking for something along these lines:

Definition
For $n\in\mathbb{N}$, a path $p$ is $n$-piecewise simple iff it is the finite concatenation of $n$ simple subpaths, that is iff there are $x_0<...<x_n\in [0,1]$ such that $x_0=0, x_n=1$ and $p$ is injective on the interval $[x_i, x_{i+1}]$ for every $i< n$. We write $p^i$ for the subpath obtained from $p$ restricted to $[x_i, x_{i+1}]$, and we write $_np$ for the set $\{x_0,....x_n\}$ of division points of $p$.

Two simple paths $p, q$ are called path-equivalent iff $p(I)=q(I)$ and $p(0)=q(0), p(1)=q(1)$ (and this definition transfers to simple subpaths).

Two $n$-piecewise simple paths $p, q$ are called strictly piecewise path-equivalent iff for all $i<n$ the subpaths $p^i, q^i$ are path-equivalent.

Finally, two regular paths $p, q$ are called path-equivalent iff there are two sequences of $n$-piecewise simple paths $(p_n), (q_n)$ ($n\in\mathbb{N}$) such that

(i) $p, q$ are the uniform limit of $(p_n), (q_n)$ respectively
(ii) $p_n, q_n$ are strictly piecewise path-equivalent for all $n$
(iii) $_np_n\subset$ $_{n+1}p_{n+1}$ and $_nq_n\subset$ $_{n+1}q_{n+1}$ for all $n$

Of course this raises two new questions :-)

1. Is each regular path the uniform limit of a sequence of piecewise simple paths with nested division-point sets?
2. If $p, q$ are two regular path-equivalent paths, is there some monotone increasing self-homeomorphism $\phi$ of $[0,1]$ such that $p=q∘ϕ$?

But I leave it at this for now, and await reactions and answers before tackling these questions myself.

Because regular paths can be very wild indeed, it seems to me that Andreas' answer in the comments above is in fact the most intuitive definition...! (call two regular paths $p, q$ path-equivalent iff $p=q∘ϕ$ for some monotone increasing self-homeomorphism $\phi$ of $[0,1]$).

But from your (the OP's) comments I gather that you would prefer a definition that focuses more (also) on the path-images. Perhaps you're looking for something along these lines:

Definition
For $n\in\mathbb{N}$, a path $p$ is $n$-piecewise simple iff it is the finite concatenation of $n$ simple subpaths, that is iff there are $x_0<...<x_n\in [0,1]$ such that $x_0=0, x_n=1$ and $p$ is injective on the interval $[x_i, x_{i+1}]$ for every $i< n$. We write $p^i$ for the subpath obtained from $p$ restricted to $[x_i, x_{i+1}]$, and we write $_np$ for the set $\{x_0,...,x_n\}$ of division points of $p$.

Two simple paths $p, q$ are called path-equivalent iff $p(I)=q(I)$ and $p(0)=q(0), p(1)=q(1)$ (and this definition transfers to simple subpaths).

Two $n$-piecewise simple paths $p, q$ are called strictly piecewise path-equivalent iff for all $i<n$ the subpaths $p^i, q^i$ are path-equivalent.

Finally, two regular paths $p, q$ are called path-equivalent iff there are two sequences of $n$-piecewise simple paths $(p_n), (q_n)$ ($n\in\mathbb{N}$) such that

(i) $p, q$ are the uniform limit of $(p_n), (q_n)$ respectively
(ii) $p_n, q_n$ are strictly piecewise path-equivalent for all $n$
(iii) $_np_n\subset$ $_{n+1}p_{n+1}$ and $_nq_n\subset$ $_{n+1}q_{n+1}$ for all $n$

Of course this raises two new questions :-)

1. Is each regular path the uniform limit of a sequence of piecewise simple paths with nested division-point sets?
2. If $p, q$ are two regular path-equivalent paths, is there some monotone increasing self-homeomorphism $\phi$ of $[0,1]$ such that $p=q∘ϕ$?

But I leave it at this for now, and await reactions and answers before tackling these questions myself.