Better approximations with two sticks

How approximation ought to be done always confuses me. Considering this problem in a textbook:

Two massless sticks of length $2r$, each with a mass $m$ fixed at its middle, are hinged at an end. One stands on top of the other. The bottom end of the lower stick is hinged at the ground. They are held such that the lower stick is vertical, and the upper one is tilted at a small angle $ε$ with respect to the vertical. They are then released. At this instant, what are the angular accelerations of the two sticks? $ε$ is very small.

Short vision: It seemed to me, in the solution, the author (David Morin) has done approximation to the Lagrange with first order approximation to the Kinetic energy term, and then second order approximation to the potential energy term.

I have two questions, and listed them at the bottom of this post.

Long vision: the kinetic energy of the Lagrangian explicitly is:

$$frac{1}{2}mr^2[(2dottheta_1costheta_1-dottheta_2costheta_2)^2+(-2dottheta_1sintheta_1-dottheta_2sintheta_2)^2]+frac{1}{2}mr^2dottheta_1^2$$

To my understanding,

With 1st order approximation $^{dagger}$, we obtain:

$$frac{1}{2}mr^2(2dot theta_1 − dot theta_2)^2+frac{1}{2}mr^2dottheta_1^2$$

The potential energy is:

$$mgr(3costheta_1 + costheta_2)$$

with 2nd order approximation:

$$mgr(4− frac{3}{2}θ_1^2 − frac{1}{2}θ_2^2) $$

Hence obtain EOMs:
$$5ddottheta_1 − 2ddottheta_2 = frac{3g}{r}theta_1$$
$$-2ddottheta_1 + 1ddottheta_2 = frac{g}{r}theta_1$$

With the initial conditions $theta_1=0$ and $theta_2=varepsilon$

$$ddot θ_1=frac{2gvarepsilon}{r}$$
$$ddot θ_2=frac{5gvarepsilon}{r}$$

I used a different approach of approximation — I used the small angle approximation right at the beginning, and assuming at the very instant the upper stick just behaved like a mass $m$ on top of a long stick with length $3r$ — then obtained an answer in a same mathematical form, but with a difference by one order of magnitude, ($ddot theta_1 = gvarepsilon/3r$).

Now, I am so profoundly confused in two sense: one technically, another methodically.

1. Technically, can we make different orders-approximations in one single equation?
2. Methodically, how do I know which one is a better approximation? (In other words, how to make better approximation?)

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$dagger$ He explained:

We can simplify this, using the small-angle approximations. The terms involving sin θ are fourth order in the small θ’s, so we can neglect them. Also, we can approximate cos θ by 1, because this entails dropping only terms of at least fourth order. So the top mass’s kinetic energy becomes $frac{1}{2}mr^2(2dottheta_1 − dottheta_2)^2$