Were I to define evil, I would say it is as you defined it. Human and animal suffering is separate from evil and I do not believe God is off the hook for all human and animal suffering, I but I do believe God us off the hook for all evil acts humans commit which are either unjustified or intended on purpose because I believe that humans have free will to choose.
As the question of good and evil emanating from the same God was always a puzzle to me, I made it my goal to sort this dilemma out for myself. Here is what I discovered through my study of the Bible.....FWIW.
In nature, and in everything that pertains to human activity, there are always equal opposites.....its the balance of nature.....pure and simple..."for every action, there is an equal and opposite reaction"....we take that for granted....in, out....up, down....forwards, backwards....with, without.... If good exists, then evil must of necessity be its equal opposite...a natural balance.
Going back to the garden of Eden, what do we find? We find God shielding humankind from a knowledge of good and evil....not from just knowing about it, but from experiencing it. The tree that humans were not to touch was placed in the garden as God's property. Yet because he created humans to be free willed, he had to place it there as a choice.....but in front of it was a huge deterrent...the death penalty. The only way to shield his human children from evil was for them to allow God (of their own free will) to determine what was good and what was not. It would have meant that only good things would come into the realms of their experience, whilst God kept evil away...like any good parent, his goal was to protect them.
The story goes that a rebel spirit presented the woman with a lie that would 1) take the penalty away (in her mind at least) and 2) make the eating (theft) of that fruit appear to be very attractive....she would "be like God, knowing good and evil" for herself.....it was too good to pass up, and not content, she persuaded her husband to disobey God's command as well, and guess what? Eating that fruit changed their lives forever....but it didn't make them "like God" at all. It caused evil thoughts to enter their minds, and their first evil realization was their shame at being naked. Sex has caused shame ever since, but it was never supposed to.
Their naked bodies now made them feel uncomfortable....and they responded by covering up their reproductive parts with fig leaves. Apparently that was not good enough, so God clothed the pair with long garments of skin, not only to protect their modesty, but also to protect their bodies from the thorns and thistles that the cursed ground outside the garden would now grow for them.
They went from eating ripe lush fruit for the picking, to having to till the unproductive soil to produce crops and make bread.....eking out a difficult existence requiring sweat and hard work.
It also meant that their bodies, (now 'unplugged' from the Creator and the tree of life that was at their disposal in the garden,) would begin to age, succumb to sickness and die....something that they were not designed for.
Within one generation a murderer was produced as a knowledge of evil became the first act of evil.....and the rest, as they say, is history....we have been struggling with the fruitage of sin and disobedience ever since....evil acts continued to invade human activity.
This is the greatest object lesson that humans will ever experience. They chose to step outside of God's protection to decide for themselves what was good and what was evil, but imperfection was to daunt their every step. Nothing they had in mind to do would prove to be successful, because now God was not protecting them. Selfishness caused this problem and selfishness continues to dominate in the world of their making. We are seeing what independence from God looks like. Its not pretty.
So from my perspective, it makes complete sense that good and evil exist in this world, and God has promised to end this object lesson and bring us back to what he started....with lessons learned and precedents set for all time to come. No one will ever be able to promote independence from God ever again....and he can get on with whatever he has planned for this vast Universe.....and he has forever to do it.