In laymans terms, what exactly is a "One time pad"? Background

I am an undergraduate novice learning about cryptography. Recently my professor introduced us to the concept of a "One time pad".

According to my professor: 


  A one time pad is a cipher that exists when one unique key is used
  once to encipher/decipher one message. It is also the only genuinely
  unbreakable cipher.


However, this simple definition seems a bit anti-climatic for describing the world's "best possible cipher" - and has left me with some lingering questions.

Questions

1) Does that mean that if I encrypt the message Hello World with the key Password123 - that this would be considered a one-time-pad?

2) What if I encrypt two different messages Hello World 1 and Hello World 2 with the same key Password123 - is this no longer a one-time-pad?

3) What if I repeat the previous scenario with a small twist. Still encrypt two different messages with the same key, but introduce a salt Password123 + [SALT] - are we now back to being a one-time-pad? (keep in mind, I am assuming the salt is sent alongside the encrypted text).

4) Does the strength of the cipher even matter as long as one unique key is used once to encipher/decipher one message. Would a one-time-pad using AES have the same strength as a one-time-pad using ROT?

So in terminology that a novice or layman like me can understand. What exactly does it mean for a cipher to be a "One time pad"?
I'll answer your questions (1, ..., 4) first:


  1) Does that mean that if I encrypt the message Hello World with the key Password123 - that this would be considered a one-time-pad?


No. The one time pad (OTP) requires the key to be truly random - otherwise it's guessable. And encrypt is vague - this could be any cipher.


  2) What if I encrypt two different messages Hello World 1 and Hello World 2 with the same key Password123` - is this no longer a one-time-pad?


No - this is called a multi time pad and is crackable. And the OTP needs the key to be the same length as the message.


  3) What if I repeat the previous scenario with a small twist. Still encrypt two different messages with the same key, but introduce a salt Password123 + [SALT] - are we now back to being a one-time-pad? (keep in mind, I am assuming the salt is sent alongside the encrypted text).


The OTP doesn't take a salt, though I suppose what you suggest is plausible. If the salt is different for each message, than that avoids multi-time pads, but sending it with the message isn't particularly secure. Like sending half a password. Plus you still have the plaintext-key length problem.


  Does the strength of the cipher even matter as long as one unique key is used once to encipher/decipher one message. Would a one-time-pad using AES have the same strength as a one-time-pad using ROT?


The OTP is a cipher$^1$ in itself. You don't use AES or ROT. And that voids the rest of the question.

The original OTP was based of the Vignère cipher. In Vignère, the key fred with message plaintext means that the key isn't long enough. Therefore, it repeats and becomes:fredfredf. Then, it's encrypted with a vigénere square, and you have the ciphertext. 

The old OTP is almost identical; the key must be random though. Truly random. And the same length as the ciphertext.

The modern OTP is much simpler:

$C_i = (P_i \oplus K_i)$

Now you might wonder: that's so simple! Why don't we use it all the time? It is, after all, unbreakable.

The answer is because of its keysize. For a 5GB message (or file) you need a 5GB key. See the problem?

As an afternote, read Crypto101 (https://crypto101.io). It's a work in progress, (some sections simply contain TODO) but it'll be beneficial for anyone interested in cryptography. 

$1$: The one time pad's status as a "cipher" is argued about all the time. I use that term here because it gets the meaning across.