Question from Katok and Hasselblat (page 724, proof of the Poincare Lemma)

Question

I was reading about the proof of the Poincare lemma. The authors define the (time dependent) vector field $X_t(x):= x/t$ and claim that the flow of it (which I think by flow they mean isotopy) is $\varphi_t(x) = tx$ and is defined for $t > 0$.

Next they argue that since $d\omega = 0$,

$$d/dt(\varphi^t)^*\omega = (\varphi^t)^*\mathcal{L}_{X_t}\omega = (\varphi^t)^*d(\omega(X_t, .)) = d((\varphi^t)^*\omega(X_t, .)).$$

They finish by saying

$$\omega - (\varphi^{t_0})^*\omega = (\varphi^1)^*\omega - (\varphi^{t_0})^*\omega = d\int_{t_0}^1(\varphi^t)^*(\omega(X_t, .))dt.$$

But $$\lim_{t_0\rightarrow 0} d\int_{t_0}^1(\varphi^t)^*(\omega(X_t, .))dt= d\int_0^1(\varphi^t)^*(\omega(X_t, .))dt$$

as $(\varphi^t)^*\omega(X_t(x), .)(v) = \omega_{tx}(x/t, .)(tv) = \omega_{tx}(x, v)$.

Here it seems that they finish the proof but I still don't understand what happens to $\omega - (\varphi^{t_0})^*\omega$ when $t_0 \rightarrow 0$ for that $\varphi^0$ is never defined in the first place. My confusion lies here. When I want to work with isotopies, I think of a family of diffeomorphisms $f_t$ where $t$ varies in a closed interval $[a, b]$. (By composing every family member we can get the isotopy $\varphi^t := f_t\circ f_a^{-1}$ with $\varphi^0 = \mathrm{id}$.) In the above argument however, there is no closed intervals involved. I have two ideas in my mind about what might going on but I'm not sure.

1. Do the authors mean that first take $\epsilon > 0$ and consider the family diffeomorphisms $\varphi^t(x) := tx$ where $t \in [\epsilon, 1]$, then consider the isotopy $\varphi^t \circ (\varphi^\epsilon)^{-1}$, do their computations to get some integral equality and then let $\epsilon \rightarrow 0$? Of course $\varphi^0 (x) = 0.x = 0$ and therefore $(\varphi^0)^* \omega = 0$ but it I don't understand how $\varphi^0$ is related to $X_t$.

2. The number $1$ is actually playing the role of $0$ in the definition of isotopy, i.e. instead of asking for $\varphi^0 = \mathrm{id}$, we are asking for $\varphi^1 = \mathrm{id}$ to hold.

I also went through a proof in WikiPedia (https://en.wikipedia.org/wiki/Poincar%C3%A9_lemma#Proof_using_Lie_derivatives) where again, I have problem with the fact that they are trying to compute $\left.\dfrac{d}{ds}g_s\right|_{s = 0}$, $g_0$ is not defined (since $f_0^{-1}$ is not defined) but in the definition of derivative, we need $g_0$.