I think we can divide the cherishing of oneself and others into four levels.
First Level: Unintelligent Self-cherishing.
So in this level the person is mainly concerned with her own happiness and with her own non suffering. Because all beings wish to be happy and not suffer, including animals.
So for this person what matters most is her own happiness even if for that she has to harm others. So this person doesn't care about the happiness of others nor the non suffering of others. Of course, the degree of this selfishness will vary regarding each person, but in a general view it functions like this.
This is a big problem because this person's thoughts will be polluted and she will commit much more non virtuous actions of body, speech and thought, which as a consequence will make this person to be born in the lower realms of samsara which are the hell realms, hungry ghost realm and animal realm, for many many years.
Second Level: Intelligent Self-cherishing.
So this person has more merit and wisdom than the first one, still this person is also mainly concerned with her own happiness and her non suffering, yet as this person believes in the law of karma, she knows that if she does good things to others the result will be her own happiness and her non suffering. So because this person wants her own happiness, she does good things to others.
Third Level: Equalizing Self and Others.
So this person understands that her happiness and the happiness of all other sentient beings has the same importance. Because all sentient beings without exception want to be happy and not suffer. So in this sense all sentient are the same. So through this reasoning this person considers her own happiness and suffering as a important as the others happiness and suffering. So through this, the person reduces her self-cherishing and as a result her negative thoughts of anger, jealousy, pride, and so on are reduced while her good thoughts like thoughts of compassion, love, wisdom, etc, become more prominent. So the result of this is the increasing of her own and others happiness and the reducing of her own and others suffering.
The Fourth and Final Level: Exchanging Self for Others.
So this person has accumulated a lot of merit in the past and also has great wisdom. So she understands that the self-cherishing thought is the root of all suffering, because all the negative karma we do is due to cherishing ourselves, the self-cherishing thought is the main reason why we all do harm to others. We do harm to others because we consider our happiness and our non suffering more important than the others happiness and others non suffering, and this arises due to our self-cherishing. Also this person understands that all her happiness comes from others, because all our happiness arises as a result of the good karma we accumulated in the past and the only way for us to do good karma is by doing good things to others.
So the reasoning here is that what we call "self" is the root of all suffering, and what we call "other" is the source of all happiness. So as a result, this person exchanges self and other by rejecting the self-cherishing and cherishing others instead. So this person trains in cherishing others more than herself. As a result the negative thoughts of this person lose even more power and the positive thoughts increase even more. This person is able to bring a lot of happiness to others by helping them selflessly, and as a result, all her good is accomplished as a secondary effect.
And these are the four levels of cherishing oneself and others which consider to exist.