If you want an expression that is similar to "leave a trace" which means "working hard to help people so I make them remember me in beautiful moments", then you could say "I want to make an impression" or "I want to make a good impression on her" or "I want to leave him with a good impression of me". 'Impression' has a similar literal meaning to 'trace', but you can only use 'impression' in the above expressions.

No. "A trace" can mean "a tiny amount." "There was a trace of lemon in the tea." It can mean that you can tell that something was once there, because it left a trace behind. "I tried to erase the pencil mark completely. It is almost gone. However, if you look carefully you can still see a faint trace of the mark." "Trace" does not mean "memories." However, you could certainly say that peoples' memories of you are "traces I have left behind." And you could certainly say "I hope I have made a good impression, and that years from now I will have left traces in their memories."

No, the sentence "leave a trace" does not have the same meaning as "working hard to help people so I make them remember me in beautiful moments." "Leave a trace" simply means to leave a mark or evidence of your presence or actions, but it doesn't necessarily imply anything about helping people or creating beautiful memories. It can be a more general statement about making an impact or being remembered in some way. On the other hand, "working hard to help people so I make them remember me in beautiful moments" specifically focuses on putting in effort to assist others and creating positive memories for them to remember you by. It has a more specific and personal connotation compared to the more general idea of "leave a trace."

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