I drive a white van for a living, something pretty similar happened to me last week, like you, I was frustrated and feeling impatient, and just did not want to let anyone in, now, when the situation is reversed, and I am the one trying to merge, and someone else makes it clear they are not going to let me, I accept it, and wait for them to pass, I expect other road users to follow this, in fact, I normally try and make eye contact with the driver, and if it is obvious that they are hell bent on not allowing me to merge, its pointless forcing the issue.
However, whilst on the main road, this white brand new Volvo estate merging from the side, insisted on nudging his nose and making me brake hard so that he could get a foothold in front of me, no eye contact, just a determined look on his face that he was entitled to do that, I wound my window down and called him a wanker, he gave me two fingers, I laughed back at him, he laughed back at me, and that was pretty much the end of it.
You were just letting off steam, it happens, don't beat yourself up over it, on a normal day, with no kids in the back, you would have reacted differently, it certainly does not define you, any more than what I did defines me, no one got hurt, no damage was done, sometimes even us humans need to bark at each other like dogs, in this world of fake civility, we sometimes forget that