An apology can become an empty exchange when the only goal is to make a child say sorry quickly. The more useful questions are what happened, who was affected and what can be done now. Words matter, but they work best when connected to an action the child understands.
First stop any ongoing harm and support the person who was hurt or upset. Do not make the injured child wait while the adult negotiates the perfect apology. Give the child who caused the problem time to settle enough to understand a brief conversation.
You knocked down the tower your brother was building is more concrete than you were mean. Help the child recognize the effect without assigning a fixed label. Depending on their age, you might offer words they can use or ask how they would explain what happened.
A repair could involve helping rebuild, returning an item, cleaning a mess or giving someone space. Ask what would actually help where appropriate, while recognizing that the affected person may not want immediate interaction. They do not have to accept a hug or declare everything fine to complete the adult's preferred ending.
Do not require physical affection as proof of remorse. A spoken apology, a note or a practical repair may be suitable, depending on the people and situation. The child can learn that the other person's boundary still matters after an apology.
When you make a mistake, name it plainly. I interrupted you and did not let you finish is clearer than sorry if you were upset. Say what you will do differently and then try to follow through. Avoid asking the child to reassure you that you are still a good parent.
For repeated behavior, an apology alone is not the whole response. Review the boundary, supervision or arrangement that needs to change. A child should not learn that saying the right word automatically restores access to a toy being used unsafely.
Keep the process proportionate. An ordinary disagreement may need a short repair, not a family hearing. The aim is to connect responsibility with understanding and action, while leaving room for people to remain upset and recover in their own time.