Let $x$ be a point of topological space $X$, and let $\omega$, $\tilde{\omega}$ be two loops based at $x$; $\omega, \tilde{\omega}: I \to X$, with $I$ being the closed interval $[0,1]$. We say that $\omega$ and $\tilde{\omega}$ are homotopic if there exists a continuous function $H: I\times I \to X$, with certain properties: at the boundary points of the second factor $I$, and for any value $s\in I$ (the first factor) $H_s(0)=H_s(1)=x$; vice versa for the $I$ spaces $H_0(t)=\omega(t)$ and $H_1(t)=\tilde{\omega}(t)$ for all $t\in I$. Most notably, being homotopic is an equivalence relation.
Is it necessary that $I$ be closed? Can I define a homotopy $H$ even for non closed $I$?
It seems that via a function $H$ I can always define an equivalence of curves. Can I define, for example, the tangent bundle of a manifold in the same way? Namely, declare $\gamma , \tilde{\gamma}:\mathbb{R}\to M$ as equivalent if there exists, a homotopy type function, $H:\mathbb{R}\times U \to M$, where $U$ labels the family of curves. Can I endow $H$ with certain properties to make sure that $\gamma \sim \tilde{\gamma} \ \Leftrightarrow \ \gamma(0)=\tilde{\gamma}(0) \ \text{and} \ \left.\dfrac{d\gamma^\mu}{d\epsilon}\right|_{\epsilon=0}=\left.\dfrac{d\tilde{\gamma}^\mu}{d\epsilon}\right|_{\epsilon=0}$?
The concept of homotopy of paths extends to non-closed paths. In fact, for $x, y \in X$ let $P(x,y)$ the set of all paths $\omega : I \to X$ such that $\omega(0) = x, \omega(1) = y$. Then a homotopy of paths from $\omega \in P(x,y)$ to $\omega' \in P(x,y)$ is map $H : I \times I \to X$ such that $H(t,0) = \omega(t), H_1(t) = \omega'(t)$ and $H(0,s) = x, H(1,s) = y$ for all $t,s$. In other words, $H$ is required to be a homotopy rel. $\{ 0, 1 \}$.
For curves defined on a non-closed interval $J$, e.g. $J = \mathbb R$, you have the problem that there are no endpoints which can be kept fixed under homotopies. Of course you can consider a subset $A \subset J$, two curves $\omega, \omega' : J \to X$ such that $\omega \mid_A = \omega' \mid_A$ and homotopies rel. $A$ (i.e. being stationary on $A$), but this has some arbitraryness. In your question you may take $A =\{ 0 \} $, but it would not make much sense to take a bigger $A$. Now you have the problem that all curves agreeing at $0$ are homotopic. Of course can additionally require that the homotopy is smooth and keeps derivatives at $0$ fixed. But I can't see any advantage in doing so. It is better to define $$\gamma \sim \tilde{\gamma} \ \Leftrightarrow \ \gamma(0)=\tilde{\gamma}(0) \ \text{and} \ \left.\dfrac{d\gamma^\mu}{d\epsilon}\right|_{\epsilon=0}=\left.\dfrac{d\tilde{\gamma}^\mu}{d\epsilon}\right|_{\epsilon=0} .$$