How do you introduce your nickname in Korean?
For example, if I want to tell the person to call me a nickname in my native language:
"My name is____, but please call me ______"

Also, If I want to introduce myself in my Korean name, like
"My name is ______, but my Korean name is ______"

Question

How do you introduce your nickname in Korean?
For example, if I want to tell the person to call me a nickname in my native language:
"My name is____, but please call me ______"

Also, If I want to introduce myself in my Korean name, like
"My name is ______, but my Korean name is ______"

Answer

"My name is____, but please call me ______" 
내 이름은 _____인데, '미키라고/노엘이라고'  부르면 돼.
* 미키 has not a final consonant(Mickey ends with a vowel). Therefore, 미키 + 라고.
  노엘 has a final consonant(Noel ends with a consonant). Therefore, 노엘 + '이'라고.

"My name is ______, but my Korean name is ______"
내 미국 이름은 ______인데(이고 is also acceptable.), 한국 이름은 ________야/이야.
* 영수 + '야'
   영숙 + '이야' (The word form changes because of the final consonant in the preceding syllable 숙. By adding '이' vowel between '영숙' and '야', 영숙이야 incorporates a more comfortable chain of sounds  for the speaker and the listener.

Answer

"My name is AA, but please call me BB" => 제 이름은 AA예요. 하지만 BB라고 ㅈ[BB라, BB로] 불러 주세요.

"My name is AA, but my Korean name is BB" => 제 이름은 AA(이)고 한국어 이름은 BB이에요[예요]".
(If the name ends with a consonant, 이고/이에요, otherwise 고/예요. 영숙이고/영숙이에요, 철수고/철수예요).

* Nickname is called 별명: 제 별명은 슈퍼맨이에요 (My nickname is Superman).
* There is also 애칭(pet name): 제 이름은 이정희, 애칭은 희야예요 (My name is Lee Jeong Hee, and pet name is Heeya).
* Korean name = 한국어 이름(or 한국 이름); English name = 영어 이름;  American name = 미국 이름.