Let us for simplicity address OP's question in the context of point mechanics where $q^i$ are generalized position coordinates on a manifold $M$ [instead of considering field theory with fields $\phi^{\alpha}(x)$]. OP's question is rooted in the difference between

1. on one hand, an infinitesimal variation $$\tag{1} q^i~\rightarrow \widetilde{q}^i~=~ q^i+\delta q^i $$ of the generalized position coordinates, or equivalently, $$ \tag{2} \delta q^ ~:=~ \widetilde{q}^i-q^i; $$

2. and on the other hand, that of a generator/Lie algebra element/vector field $$ \tag{3} Y~=~Y^i\frac{\partial}{\partial q^i},$$ which is not infinitesimal (although $Y$ is sometimes confusingly referred to as an 'infinitesimal generator' in the literature).

Both concepts $\delta$ and $Y$ are linear derivations that satisfy Leibniz rule, and the interrelation between the two is given by

$$ \tag{4} \delta q^i~=~\epsilon Y^i, $$

where $\epsilon$ is an infinitesimal parameter. [We should mention for completeness that the mathematical concept of a vector field is intimately tied to the concept of a flow.]

The (bare) Noether charge

$$ \tag{5} Q ~=~p_i Y^i$$

is (in this case) momentum

$$ \tag{6} p_i ~:= ~\frac{\partial L}{\partial \dot{q}^i} $$

times generator $Y^i$. In particular, the definition (5) of the Noether charge $Q$ does not depend on the $\epsilon$ parameter.

I) Let us for simplicity address OP's question in the context of point mechanics where $q^i$ are generalized position coordinates on some manifold $M$ [instead of considering field theory with fields $\phi^{\alpha}(x)$]. OP's question is rooted in the difference between

1. on one hand, an infinitesimal variation $$\tag{1} q^i~\rightarrow \widetilde{q}^i~=~ q^i+\delta q^i $$ of the generalized position coordinates, or equivalently, $$ \tag{2} \delta q^ ~:=~ \widetilde{q}^i-q^i; $$

2. and on the other hand, that of a generator/Lie algebra element/vector field $$ \tag{3} Y~=~Y^i\frac{\partial}{\partial q^i},\qquad Y^i~=~Y^i(q),$$ which is not infinitesimal (although $Y$ is sometimes confusingly referred to as an 'infinitesimal generator' in the literature).

Both concepts $\delta$ and $Y$ are linear derivations that satisfy Leibniz rule, and the interrelation between the two is given by

$$ \tag{4} \delta q^i~=~\epsilon Y^i, $$

where $\epsilon$ in eq. (4) is an infinitesimal parameter. The mathematical concept of a vector field $Y$ is tied in a bijective manner to the concept of a flow$^1$

$$ \tag{5} \sigma:~]\!-\!c,c[ ~\times~ M~\to~ M, \qquad ]\!-\!c,c[ ~\subseteq~ \mathbb{R},$$

where

$$ \tag{6} \frac{d}{d\epsilon}\sigma^i(\epsilon,q)~=~Y^i(\sigma(\epsilon,q)), \qquad\sigma^i(\epsilon=0,q)~=~q^i. $$ A flow $\sigma$ satisfies $$ \tag{7} \sigma^i(\epsilon,\sigma(\epsilon^{\prime},q))~=~\sigma^i(\epsilon+\epsilon^{\prime},q).$$ Note that in eq. (7), it is understood that $\epsilon$ and $\epsilon^{\prime}$ are real numbers in the interval $ ]\!-\!\frac{c}{2},\frac{c}{2}[ \subseteq \mathbb{R}$, and not infinitesimal.

II) The (bare) Noether charge

$$ \tag{8} Q ~=~p_i Y^i$$

is (in this case) momentum

$$ \tag{9} p_i ~:= ~\frac{\partial L}{\partial \dot{q}^i} $$

times generator $Y^i$. In particular, the definition (5) of the Noether charge $Q$ does not depend on the $\epsilon$ parameter.

--

$^1$ We ignore the possibility that the domain $]\!-\!c,c[$ could depend on the initial position $q\in M$.