A little clarification. You would use:
- Witam or Witaj when welcoming a single person
- Witam or Witajcie when welcoming a group of people
- Witamy or Witaj when you and sb else (at least one more person) welcomed a single person
- Witamy or Witajcie when you and sb else (at least one more person) welcomed a group of people
The first two – yes, absolutely, that is the literal meaning.
The second ones – might be, but only when welcoming someone eg. to your house, not as a response after being thanked.
Because witaj and witajcie are in imperative mood (2nd singular and 2nd plural persons), both translate literally to English as just welcome!.
To be honest, "witam" implies that you are more important than the one you're talking to. It's usually quite subtle, but sounds really bad in e.g. formal letters ("witaj" also doesn't sound well). "Witam" is adequate when you invite somebody to your house - you can tell them "witam".
I read a lengthy article on a polish grammar board about that (need to find that one again) and asked my cousins after and all tend to disagree on the "more important" notion. The conclusion was that only a really small majority (sort of grammar nazis that try to preserve the old ways) still holds on to that notion but that it's pretty much outdated in Poland, even in a formal context. Just thought to add that here.
"Grammar nazis" in Poland are called "professors" and they all tend to agree that "witam" implies your importance. The fact that some users do not see the difference does not mean that the difference does not exist. Of course, it's a subtle way, and perfectly fine if you're talking to your classmate, but it is inappropriate while talking with your boss.
Your "All tend to agree" statement is hardly an argument. "Grammar nazis" are not called "professors", but some of course can be called that. Most of my family is born and still lives in Poland, with my uncle having production facilities for filling plant equipment there (the engineering department is located in Germany) and one of my cousins is in the process of finishing his doctoral program in Wrocław. None of them agrees with you nor does the majority of posts I read so far - I keep searching for more of course. At the end of the day, what do I know, right?! It's probably better to be save than sorry even though it seems inaccurate to state it as clear cut as you do apparently. With this in mind, witam! ;D
I never said that "rule" does not exist (though I'd be interested in an official link to it of your so called Language Council if you have the time) nor that you will have a hard time finding a professor that argues for it, all I'm saying is that many don't care about it anymore in everyday life apparently. Certainly enough to mention it and that's what I did. The conclusion here: One just has to be more important than everyone else. I am certainly working on it :D
All of these expressions should only be used in spoken language. They have very significant differences in meaning and application.
Witam (I greet you)/Witamy (We greet you) are the formal, noble,
and polite greetings used from the position of the host(s) of the conference, TV/radio show or from the position of the head of the family, or any other person in authority. These expressions refer to the public, or invited guests, who are not family or close friends yet.
Witaj (Happy to see you-one person)/ Witajcie (Happy to see you-
two or more people) are very common, casual expressions used to greet peers, younger family members, and close friends at school,
at the party, and other fun social events. These expressions have reappeared in spoken language as quite convenient one-word equivalents of the word welcome and they've become so common, they've actually reduced the meaning of this word to the simple greeting Witaj/ Cześć/ Hej (Welcome/ Hello/ Hi).
"Welcome!" is much more than a simple greeting. It is a joyful, warm and inviting way to say: "I am happy to see you!/ Nice to see you!/ Come over!" when you open your arms and greet your friends at the door of your house.
The words "Witaj!/Witajcie!" (Hello!/Hi!) used commonly as equivalents of
the "Welcome!" might be short, easy and convenient to use, but they do not reflect the full meaning of this word. The expressions Cieszę się, że cię
widzę! (Witaj!)/ Cieszę się, że was widzę (Witajcie!) are much better...
Anyone who speaks a romanic language could help me with a doubt? "Witaj" is used like "benvenuto", "bem-vindo" and "bienvenido" (example: someone arrives at your home and you say "witaj"), or it's a reply for a thanks, like "prego", "de nada" and "por nada"? (Example: you did a favor to someone, the person thanks me and I reply with "witaj")