Hereis a part of the text from one of the links you provided.
The true meaning of “As-Salam Alaikum” is not a mere “Hi”, “Hello”, “How are you” as we are accustomed to using today – it has a much deeper meaning than just a standard greeting. In actuality, there are three meanings for this greeting:
1. As-Salam, as we know, is one of the names of Allah. Thus, when we say “As-Salam Alaikum” we are actually saying that may the trait of Allah (as-Salam or peace and tranquility) be upon you and may He protect you;
2. As-Salam is also in the meaning of submission or surrender. Thus, when we say “As-Salam Alaikum” we are actually saying that we submit to what you would like for us to do (obviously within the limits of the Shariah.);
3. As-Salam is also in the meaning of protection or safety. Thus in this meaning, when we greet another believer with “As-Salam Alaikum” we are actually guaranteeing our believing brother or sister protection from any evil from ourselves and that we will not do a single thing to harm them – either physically or even spiritually. Not only would we not harm them with our hands, but we will also not cause them grief with our tongue...
This is what i was referring too when i said it doesnt just mean simply "Peace be upon you"
peace.
The true meaning of “As-Salam Alaikum” is not a mere “Hi”, “Hello”, “How are you” as we are accustomed to using today – it has a much deeper meaning than just a standard greeting. In actuality, there are three meanings for this greeting:
1. As-Salam, as we know, is one of the names of Allah. Thus, when we say “As-Salam Alaikum” we are actually saying that may the trait of Allah (as-Salam or peace and tranquility) be upon you and may He protect you;
2. As-Salam is also in the meaning of submission or surrender. Thus, when we say “As-Salam Alaikum” we are actually saying that we submit to what you would like for us to do (obviously within the limits of the Shariah.);
3. As-Salam is also in the meaning of protection or safety. Thus in this meaning, when we greet another believer with “As-Salam Alaikum” we are actually guaranteeing our believing brother or sister protection from any evil from ourselves and that we will not do a single thing to harm them – either physically or even spiritually. Not only would we not harm them with our hands, but we will also not cause them grief with our tongue...
This is what i was referring too when i said it doesnt just mean simply "Peace be upon you"
peace.