simplifying expressions - Eliminate a variable from a differential equation and an algebra equation then collect the remaining variables

Recently, I am attempting to eliminate a variable from a differential equation (DE) and an algebra equation (AE), then collect the remaining variables as a new (grand) parameter. Actually, I have worked on this problem more than one week. Due to my incompetence, today I seek someone's help on this forum. To show this idea, this is a working example:

Implement the following code step by step:

eq1 = h'[t] == Log[(1 - xi)/(1 - x0)];
eq2 = h[t] == (xw - xi)/(-(1 - xw)*Log[(1 - xi)/(1 - x0)]);
Eliminate[{eq1, eq2}, xi] // FullSimplify

Then I get a DE

E^Derivative[1][h][ t] + ((-1 + xw) (-1 + h[t] Derivative[1][h][t]))/(-1 + x0) == 0

Next, I want to replace the remaining variables $\frac{1-x0}{1-xw}$ by a new parameter $c$, in other word, the desired DE is $e^{h'[t]}-\frac{1}{c}(1-h[t]h'[t])=0$, which is a preferred form for parameter study. So I tried

%/. (1 - x0)/(1 - xw) -> c

unfortunately, this replacement does not work, which follows my first question: how to replace a combination of variable by a parameter. This question appears to be silly, someone may say that why not replace it manually? The reason is in some more complicated equation a number of parameters are often distributed, so it is not easy to combine them manually. You have gotten my point:)

This is my full problem:

First, parameter $\Xi$ consists of known parameters

Ξ = (EE*((1 + B KK) Sqrt[M] Log[(1 - x0)/(1 - xw)] - 
 B Δ Subscript[Θ, ∞])^2)/(\
Δ^2*(Γ*(1 - xw)*(Log[(1 - x0)/(1 - xw)] - 

Δ/Sqrt[M]*B*Subscript[Θ, ∞]) - 
 KK*B*Subscript[Θ, ∞]));

Second, function $g_1$ and $g_2$ contain unknown variable $xi$

g1 = ((Δ*Γ*(xi - xw) + KK*Sqrt[M]*
Log[(1 - x0)/(1 - xi)])*(Subscript[Θ, ∞] - 
\Γ*(xi - xw)))/((Subscript[Θ, ∞] - 
\Γ*(xi - xw))*Δ*B - 
Sqrt[M]*(1 + KK*B)*Log[(1 - x0)/(1 - xi)]);


g2 = ((Δ*Γ*(xi - xw) + 
KK*Sqrt[M]*Log[(1 - x0)/(1 - xi)])*(1 + KK*B))/(Δ*
B*(Subscript[Θ, ∞] - 
Γ*(xi - xw)) - (1 + KK*B)*Sqrt[M]*Log[(1 - x0)/(1 - xi)]);

Third, DE and AE for $H[T]$:

eq1 = H'[T] == 1/Ξ*(EE*(Γ*(xi-xw)-B*g1))/(KK + g2);

eq2 = H[T] == (Δ*Γ*(xi - xw) + 
KK*Sqrt[M]*Log[(1 - x0)/(1 - xi)])/(Δ*

B*(Subscript[Θ, ∞] - 
Γ*(xi-xw)) -Sqrt[M]*(1 + KK*B)*Log[(1 - x0)/(1 - xi)]);

Fourth, eliminate $xi$:

Eliminate[{eq1, eq2}, xi]

Note from which my second question arises: how to speed up the long-time running of Mathematica?

Finally, the last question is how to combine those parameter(known) in the above resulting differential equation into the following two new (grand) parameters:

Σ = ((Δ*Γ*Sqrt[M]*(1 - xw) -
KK*M*(1 + B KK) )*Log[(1 - x0)/(1 - xw)]^2)/((1 + B KK) Sqrt[M]*

Log[(1 - x0)/(1 - xw)] - 
B Δ Subscript[Θ, ∞])^2;

and

Π = (Δ^2*B*
Subscript[Θ, ∞]*Γ*(1 - xw)*
Log[(1 - x0)/(1 - xw)])/((1 + B KK) Sqrt[M] Log[(1 - x0)/(1 - xw)] - 
B Δ Subscript[Θ, ∞])^2;

Some Hints: the final DE might contain the following terms $H[T]$, $H'[T]$, $\Sigma$, $\Pi$, and the exponential function.

I am so sorry for the complicated expressions. And many thanks for your time!