Both are equally harassing or killing animals. However, it seems difficult to see the difference between direct and indirect. Eating
meat made by killing animals by other people is also considered an act
of participating in the livestock industry.
No it's not.
In one case you're doing active (killing) or even senseless/unnecessary harm (harassing) to a living being that is able to perceive and suffer from that harm. And in the other case you're doing things with their corpse.
Both from the perspective of the perpetrator and the victim these are not equal. Let's just set aside the animals for a second and think of the same situation for human beings. Then you'd compare desecration of corpses and cannibalism of deceased people to killing and murdering people. While neither is great, the latter is likely going to get you the longer sentence. Unless you actively helped to produce those corpses in which case you're again talking about the latter case primarily.
So regardless of the diminished legal status that we often have for animals, the two actions itself are not considered to be equal.
Also no it is specifically NOT CONSIDERED to be the same as killing the animals or even as an participating in the livestock industry and you might argue that this is part of the problem.
It's actually fairly easy to argue that that the actual "crime" of harming and killing the animal has happened before your involvement in the process even started and that your presence or absence in that process has no positive or negative influence on the fate of that particular animal because it was dead already before you bought the meat from the grocery store.
You might even argue that without you buying it, the meat would rot and the animal would have suffered for nothing.
Now that is only half of the story and you might just as well argue truthfully that buying the product acts as proof to the "producers" (those that do the killing) that there is an acceptance of and a market for their good and thus encourages them to restock the supply. So while you're not responsible for what's on display right now the fact that it sells and keeps them in business is what contributes to a partial responsibility for future killings of animals. So by eating meat you're specifically not CONSIDERED to act as a part of the livestock industry, but you ARE, regardless of that, a factor in it. As long as there is money to be made in that business, there'll be suppliers.
So it's actually fairly easy to push the responsibility to the producers, while not punishing them for their actions thereby keeping the conscious clean while everything stays the same. It's the production that produces the harm not the consumption (that causes, not the harm, but the production) so if you can manage to neatly separate the two it's not "your problem".
Another thing is that you can argue that killing living creatures up to humans is considered legal as long as it necessary to survive (self-defense for example) and that eating is necessary to survive. Now you can also argue against that saying that there are nowadays ways to get around doing that by killing other animal species. But then again people also start wars and get involved in ultimately pointless murder of our own species. So it's not that we are just hypocrites when it comes to animals we're also hypocrites when it comes to humans as well.
So TL;DR killing and torturing animals reveals something about the character of the person doing that, in that they don't empathize with the suffering of another living being. While using the products of such an action is able to pretend they're just making the best of that situation. It's not fully covered by logical arguments but just enough to distance oneself from the process while still taking part in it.