Honestly I don’t think the item 4) you suggest applies.
This item states that check-my-work question could be allowed when they are of the type
It seems as if step X is wrong? But it should be right because of Y, so why is this not the case?
Here, we have two competing derivations. Nowhere in either exposition does the OP identify a step X which is wrong, and justify it should be right because of Y. Rather OP is plainly asking for other users to decide which of the two methods is right.
I was one of the original closers, and I don’t remember why I voted to close. Looking at it now the question suffers from other problems. For starters, the variable $x$ of method 1 is defined as the extension of the spring whereas in method 2 the variables $x_1$ and $x_2$ are defined from the center of mass.
The extension of the spring should be (if I understand well) $x_1-x_2$, not $x_1+x_2$. This way, if both $x_1$ and $x_2$ are displaced by the same amount, the spring is not stretched or compressed. Because so little details are provided, it is impossible to verify one way or the other.
I would vote to close again as check-my-work or as unclear.
Clarifications could include the actual FBD illustrating the forces, a coordinate system from which the definition of $x_1$, $x_2$ and $x$ can easily be grasped. This way, the OP could identify steps where the parallel development of the two solutions differ in an essential way, and could then proceed to identify a principle which seems to apply in one case but not another - at least bring it closer to the spirit of the format you quote.