They sinned, and obviously did not die physically. The death was spiritual.
And, as we know from Scripture
Therefore, just as through one man sin entered into the world, and death through sin (Romans 5:12)
So, Adam and Eve did not die physically but, as God said they would, they did die and, through them death entered the world. Their sin necessitated the death of animals in order to cover them. I don't see anything in Genesis that indicates that the deaths of the animals was anything but a necessary act of God to provide clothes for Adam and Eve.
The 1 Corinthians Scripture actually confirms what the article says.
it is sown a natural body, it is raised a spiritual body. If there is a natural body, there is also a spiritual body. 45 So also it is written, “The first man, Adam, became a living soul.” The last Adam became a life-giving spirit. 46 However, the spiritual is not first, but the natural; then the spiritual. 47 The first man is from the earth, earthy; the second man is from heaven. 48 As is the earthy, so also are those who are earthy; and as is the heavenly, so also are those who are heavenly. 49 Just as we have borne the image of the earthy, we will also bear the image of the heavenly.
The physical part was created first, then God breathed the spirit into man. Since God ordained that sin would bring death, Adam sinned, remained alive physically, died spiritually, that condition was passed down to every human who would ever live, and it would take the death of Jesus Christ to eliminate spiritual death.
Actually, they did die physically, just not right then. There's no telling how much time passed from the creation until Adam/Eve ate the fruit. But at that instant, they transformed from immortal to mortal.