Whilst honing, if I already push the knife away from my hand, must I still push the knife towards my hand?

While it doesn't really matter in which direction you hone your knife, honing towards you is (a, but not the only) standard practice when you don't use the technique you pictured first, where a towel is placed under a vertically placed honing rod.

The reason is this: if you hone "freehanded", with the rod in one hand and your knife in the other, would you rather swing your blade outward toward another person, or toward you, where the honing rod grip usually has protrusions that stop the blade in case you slip? In a kitchen with multiple people working, blade honing outwards would endanger other people, hence why most chefs I know hone inwards.

Gordon Ramsay demonstrates the protrusions stopping a blade here, although he hones outwards, but with the back edge of the knife first.

I hone moving the honing steel because I'm a lefty. I keep the knife stationary.

To answer your question, there's no benefit, nor penalty, for honing one way, or the other. Do whatever feels natural to you.

What's important is maintaining the right angle, and also the right pressure. You should start with a high pressure and gradually you should lower the pressure.