No it really isn't, A Tsundere is Mikato from Toaru, or Chitoge from Nisekoi, or Misty from Pokemon, and half the charactors voiced by Rei Kugimiya, they have difficulty showing their love for a charactor so they end up insulting them or appearing cold and indifferent making the MC think they might not even like them. Mikato always calls Touma 'You', I don't think I ever remember her calling him by his name, Chitoge always insults the MC calling him a Beansprout, Yet these two charactors have admitted to themselves that they are in love with the main charactor. If Tsundere just meant a charactor who was hard on the MC, Index from Toaru would also be considered a Tsundere, but she's not, she's just a greedy person who likes to eat him out of house and home. Sakura has shown she loves Naruto, though it hasn't become clear if she herself has realised she is In Love with him. Nami as sworn loyalty to Luffy and would die for him, but she won't stand for his usual hairbrained schemes that will get them all in trouble (also all potential romance has been cut from the manga) and Rukia actually could be a Tsundere.
Tsun - Dere, meaning they show their love (dere) by being mean/cold/indifferent (Tsun)
Yes, it really IS. And I do not know why you have posted that paragraph when:
First - You tell a list of characters that according you are or are not tsunderes but you never explain why they are or are not tsunderes (And I have no idea of who Mikato or Chikoge are. First time that I have heard of them. But that is another matter).
Second - You put words on my mouth. When did I say that a Tsundere is a character that beats the MC?
Third - You have come up with a completely arbitrary definition in order to try to prove your argument. That is NOT what "tsundere" means and that is not what a tsundere is (and "dere" is not love, either).
From wikipedia:
"Tsundere (ツンデレ?, pronounced [tsɯndeɽe]) is a Japanese character development process that describes a person who is initially cold and even hostile towards another person before gradually showing their warm side over time. The word is derived from the terms tsun tsun (ツンツン?), meaning to turn away in disgust, and dere dere (デレデレ?) meaning to become 'lovey dovey'."
That is the official definition, and it is not what you say what it is.
The tvtropes article goes in greater depth (Although it is fulll with mistakes):
Tsundere
So yes, according the official definition Sakura IS a tsundere because she was initially hostile towards Naruto and gradually showed him her soft side.
"She would die for him but she would not stand for his hare-brained schemes" sounds like a sentence that would be used to describe most tsunderes, too. It perfectly describes Sayaka Yumi, one of the characters that defined the trope.
Your definition is completely arbitrary, and if we were to accept it instead of the official one, most tsundere characters would stop to be consider tsunderes... including the classic characters that defined the trope!
The trouble is that the trope has gone off rails through years of overexposure and misuse, so a lot of people -including anime fans in spite of they should know better- has got stuck in their heads the erroneous idea that tsundere means "Hyper-violent girl that hits the boy that she likes". That IS not what the word means and the archetype is way more complex, but that misconception generates all kind of mistakes.