This is a standard definition of regular curves. My question comes from the words "In other words". If $c:[a,b]\rightarrow M$ is an immersion, then by definition we have $$c_{*,p}:[a,b]\rightarrow T_pM$$ is injective, which by group theory means $\text{ker}\ c_{*,p}=0$, that is to say, $c'(0)=0$. But the definition says that $c'(t)$ is never zero for all $t\in[a,b]$. If $0\in[a,b]$, is this a contradiction?
The velocity is (modulo a choice of basis and notations) equal to:
$$c_{\star,p}(v)$$
where $v$ is the tangent vector of $[a,b]$ at $c^{-1}(p)$ (if you're familiar with the following notation, $v\in\mathbb{R}\frac{d}{dr}|_{c^{-1}(p)}$).
So $c_{\star,p}(v)$, for each point $p$, is a linear application from the tangent space $\mathbb{R}$ of $[a,b]$ at $c^{-1}(p)$ which has kernel $\{0\}$. If the curve is regular, the linear application itself is not zero (even if its value at the zero tangent vector is zero).
Therefore this is not a contradiction.
