I used to be short-tempered but now I'm trying to improve. Should we apologise to others for being rude? Sometimes I use loud or rude tone but later realize I've to be careful. Driver was about to start driving the car and I was holding the door to enter in the car, so I said it rudely "what has happened" now I'm thinking I should not have behaved like this and did not use such tone.

All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad is His slave and Messenger.

You have done well by striving to be forbearing, and hopefully, if you are sincere and committed, you will acquire the virtue of forbearance. A Muslim should exert effort to train himself to embody good moral qualities until they eventually become second nature (that inform his behavior without feeling any awkwardness). Good moral attributes can be acquired and learned. It was narrated by Abu Ad-Darda’ that the Prophet, sallallahu ‘alayhi wa sallam, said: “Knowledge is only acquired by learning, and forbearance is only attained by endeavoring to have forbearance.” [Al-Bayhaqi and At-Tabarani]

If your anger drove you to harm others by words or deeds, you bear sin for that, and it is incumbent to repent of it. One of the conditions for the validity of repentance is seeking the forgiveness of the wronged person. However, if you fear that it might incur any disadvantages on you, it is adequate to supplicate Allah, The Exalted, in his favor abundantly.

Allah knows best.