If I 'lied" accidentally, do I have to clarify?

If it’s really trivial matter like I didn’t raise my hand up when she asked who is going to be free after school because I was debating with myself the meaning of her question, what’s the definition of “can come”, do I have to clarify by sending her an email saying “Hey, sorry to tell you I was actually free today”? I think not, right?

But under what conditions do I have to go back and tell the truth? If, using my own judgement, telling the truth might not make a difference. Do I still have tell it anyway?

The fact that someone asks a question does not impose any obligation for you to answer it. Failure to signify an answer by raising your hand does not imply that you answer “no” (or “yes”) - it simply means you do not answer at all (and nobody can compel you to answer a question simply because it is asked of you).

Do not bring that into the question of sin.
It is a practical question: does it make a difference or not?
Do you need to clarify for the good of the group?
If it is needed, like how many go to the field trip to prepare lunches, you should say.
If does not make any difference, you do not need to say.

Your entire post is the very definition of scrupulosity. If you heard the rap sheet I take into confession, you would run screaming in horror into the night.

You are worrying about nothing.

Please do not become overly-scrupulous about this.

Let me state boldly: everybody lies at times, even those who say that they never do.
They usually don’t INTEND to, it’s often lying by omission or exxageration, almost
always VENIAL sins. If you apologize to God and really want to change, then each venial sin is forgiven, and does not heap one on top of the other, as long as you are sorry at the moment, confess it to God at least, and try to do better, even if you keep failing.

It is deliberate, schemed lying that really can get you into trouble with God.
Just ask Ananias and Sapphira.
Don’t PLAN a big lie. God really hates that. He will forgive it, if you repent,
no matter how many times you do it, if you repent each time (70 x 7 applies to him, too)
but He still hates it. He knew that Ananias and Sapphira had bad hearts and would not have truly repented, so he struck both of them dead for lying right in the face of the Holy Spirit (acts, chapter 5).

What you did is not serious at all. Don’t worry about it.
Just resolve to try and do better next time. It’s good that you have a sensitive conscience, that’s preferable to having a calloused conscience, that’s for sure.
But don’t become scrupulous where you worry yourself to death over nothing.
Move on, and God love you!!!