"I wore a dress shirt and a tie."
As a native speaker I do use the term 'dress shirt', but only in certain situations.
For example: "Should I wear a dress shirt or a polo shirt?" This shows the difference between the types of shirts I am considering.
In conversation, I will replace 'dress shirt' and use 'shirt' if it is understood or not important to the conversation. Example: "I like the color of your shirt. It goes well with the color of your tie." I wouldn't say 'dress shirt' because both you and I know you are wearing a dress shirt and the information may be omitted.
The 'ワイシャツ' is the Japanese English word. It is said that 'ワイシャツ' comes from 'white shirt' of English. We don't say 'ワイシャツ' as a casual white shirt. 'ワイシャツ' are suitable for business suits. When you search the word 'ワイシャツ' on the internet, color shirts are also appear. It means 'shirt for business' rather than 'white shirt'.
Does not that word exist in English? Is it the same thing that the word 'sister' is not distinguished to 'younger sister and 'older sister' in English?
To my knowledge (American English), "white shirt" isn't a word/term other than describing the shirt's color. Dress shirts (business/formal wear) were traditionally white in color. I can see how the Japanese term could have come from that. It is very interesting. :-) I have really enjoyed seeing which words were adopted into Japanese and how it happened.
Could it be related to "white collar jobs" (office) as opposed to "blue collar jobs" (worker)? Where the first category usually wears more formal clothes. I've seen this used in American English a fair amount, but not in British (that I can remember) I'm from Sweden so I might have just missed it, though.
https://ja.wikipedia.org/wiki/%E3%83%AF%E3%82%A4%E3%82%B7%E3%83%A3%E3%83%84 ワイシャツwiki the part '用語'.
But the truth is not known. There is a possible that your teacher is correct than wiki. Because wiki is written by somebody who we don't know. :)
(though i have heard 'ワイシャツ' means 'white shirt' before i looked at wiki. )
What "dress shirt" means in British English... https://www.collinsdictionary.com/dictionary/english/dress-shirt