The problem is that, I think the label actually throws people off in a way.
Good and Evil is
Selfless ---------------------- Selfish
Lawful and Chaotic is
Community ------------------- Individual
Good in DND terms means you value the life and dignity of others, even if you don't personally know them. You'd save some random at significant risk to yourself, just because you think an unknown actor deserves to live, until proven otherwise. Though this doesn't extend to known evil aligned races in D&D. A Paladin generally wouldn't attempt to convince a camp of Orcs to leave peacefully. This means a good character can be played as a bloodthirsty zealot in a D&D setting. A SJW may be a similar example in RL, because even if what they are doing can be perceived as evil by neutral parties, to the person in question, their actions are motivated to do good for all.
Neutral means you value people who you have personal relationships in the same way. You generally won't go out of your way to help a stranger. But at one time or another your could be convinced to be selfish or selfless, depending on the circumstances.
Evil means you will not go out of your way to risk yourself, even for those you have a personal relationship with. Unless there is something you can get out of it. This doesn't have to be material though. And evil character might do good deeds, but only because he wants the adoration of the public, the heart of a maiden, or a position appointed by the King. A sociopath for example could be evil, and yet for the most part, keep up the appearances of being an moral and upstanding member of the community.
Lawful generally means work with a community or law to achieve your goals. Whether that be getting someone wrongfully accused freed. Or manipulating public opinion or stretching the letter of the law to get someone executed. Sometimes its also themselves who are very disciplined people, and so it tends to be part of their actual personality.
Neutral is someone that will generally follow the community, but from time to time will go against the community depending on the circumstances, self perservation, greed, or moral outrage.
Chaotic means you ignore the community at large, and generally aren't interested in participating the social circle. You act, despite what the community or law may think of it. Which could cover a range of archetypes, the wanderer, the angry hermit, the vigilante, to the deranged and insane wizard.
This means it's quite easy to have Good and Evil characters to work together to reach a common goal, even thought their personal motivations may be different. But much harder for Lawful and Chaotic to work together, whereas one looks to community consensus or the letter of the law, while the other focuses on their personal interpretation and reasoning.