A Puzzle of Red Lines

One day I was doodling and I came up with a fun rule for drawing lines on a grid. Can you figure out how they work?

The 2x2 is empty. The 3x3 is impossible to construct. The 4x4 is unique (up to rotation). The 5x5 has a few variations with one or two lines moved around. The 6x6 is your mystery!

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Hint 1

This is the unique (up to rotation and reflection) solution to the 2x3 grid. I call it the elephant.

Notice how much it shows up in the other grids?

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Hint 2

These 3x3’s almost work.

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Hint 3

The gray borders are important! They have the same function as the red lines, but they’re already in place for a specific NxM grid.

The faded dark lines are merely visual indicators for the grid.

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Hint 4

The puzzle rules generalize to hex grids. Here’s a few of them.

It doesn’t seem to take as many red lines.

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Hint 5

And here’s a 3x3x3 cube from two different angles.