Various wordings for giving the greeting are mentioned in the Qur’an and Sunnah, so there is nothing wrong with the Muslim choosing whatever he wants among them. But it is preferable for him to choose the most complete and best of them so that it will increase his reward.
Giving and returning the greeting is undoubtedly one of the acts of worship encouraged by the Prophet (blessings and peace of Allah be upon him) and enjoined by him, and the ruling on acts of worship is that it is not permissible to change the wording or the manner in which they are done from that which is narrated in the texts.
The scholars of the Standing Committee for Issuing Fatwas said:
The basic principle with regard to adhkaar (pl. of dhikr – remembrance of Allah) and all other acts of worship is to limit it to that which is narrated of the phrases and the manner in which they are to be done as mentioned in the Book of Allah and the Sunnah of His Messenger (blessings and peace of Allah be upon him). End quote.
Fataawa al-Lajnah ad-Daa’imah (6/87)
Why would the Muslim turn away from the prophetic wording and the prophetic teaching in favour of something else, when the Prophet (blessings and peace of Allah be upon him) said: “The best of guidance is the guidance of Muhammad (blessings and peace of Allah be upon him).” Narrated by Muslim (867).
Allah prescribed the salaam for Adam (peace be upon him) and his descendants until the Hour begins. In fact the greeting of the believers will continue to be salaam even after they enter Paradise.
It was narrated by Abu Hurayrah (may Allah be pleased with him) from the Prophet (blessings and peace of Allah be upon him), that when Allah, may He be exalted, created Adam, He said to him: “Go and greet that group – who were a group of the angels who were sitting (nearby) – and listen to their response, for it will be your greeting and the greeting of your descendants.” So he went to them and said As-salaamu ‘alaykum (peace be upon you), and they said As-salaamu ‘alayka wa rahmat-Allah (peace be upon you and the blessing of Allah), so they added (the words) wa rahmat-Allah. So salaam is the greeting of Adam and his descendants.
Al-Bukhaari narrated in al-Adab al-Mufrad (989) that Anas (may Allah be pleased with him) said: The Messenger of Allah (blessings and peace of Allah be upon him) said: “As-salaam is one of the names of Allah, may He be exalted, and Allah has ordained it on earth, so spread (the greeting of) salaam amongst yourselves.”
Classed as hasan by al-Albaani in Saheeh al-Adab al-Mufrad (764).
With regard to the Hereafter:
Allah, may He be exalted, says (interpretation of the meaning):
“Their greeting therein will be: Salaam (peace!)”
“And those who kept their duty to their Lord (Al-Muttaqoon – the pious) will be led to Paradise in groups till when they reach it, and its gates will be opened (before their arrival for their reception) and its keepers will say: Salaamun ‘Alaykum (peace be upon you)!”
“And angels shall enter unto them from every gate (saying):
‘Salaamun ‘Alaikum (peace be upon you) for you persevered in patience! Excellent indeed is the final home!’”
[ar-Ra‘d 13:23, 24].
As that is the case, what we should do is not change the wording of this act of worship or add to it, because it is the Sunnah of the Prophets and Messengers, and it is the slogan of the believers of all nations.
With regard to the two versions mentioned in the question, the first of which is “Peace be upon you from a Lord Who is Oft-Forgiving and Most Merciful”, if this is said once or twice, there is nothing wrong with it, so long as one still believes that the prophetic wording is better. But if it is taken as a regular practice, then the least that may be said concerning it is that it is makrooh, because it is persisting in something that is contrary to the Sunnah and replacing that which is better with something that is inferior.
The aim of the Muslim when saying that should be to offer supplication for those whom he is greeting, asking Allah, may He be exalted, the Oft-Forgiving and Most Merciful Lord, to grant them peace and decree wellbeing for them.
But if his aim when saying it is to imitate the words of Allah, may He be exalted (interpretation of the meaning): “Verily, the dwellers of the Paradise, that Day, will be busy with joyful things. They and their wives will be in pleasant shade, reclining on thrones. They will have therein fruits (of all kinds) and all that they ask for. (It will be said to them): Salaam (peace be on you) — a Word from the Lord (Allaah), Most Merciful” [Yaa-Seen 36:55-58], then this is a mistaken understanding of the verse, because what this verse means is that this greeting for the people of Paradise is from Allah, may He be exalted, the Most Merciful Lord; hence it is Allah, may He be exalted, Who is greeting the people of Paradise.
See: Tafseer Ibn Katheer (3/754); Tafseer as-Sa‘di (p. 821).
The second wording mentioned in the question, “In the name of the Almighty Lord I begin my words, and upon His noble Prophet I send my salaam”, also does not come under the heading of the salaam (greeting) that it is prescribed to spread among people. It contains no greeting of people with salaam; rather it is sending salaams upon the Prophet (blessings and peace of Allah be upon him).
Moreover, the phrase “and upon His noble Prophet I send my salaam” is also subject to further discussion.
We do not say (in Arabic) “Ursilu as-salaam ila Rasool-Allah (lit. I send salaam to the Messenger of Allah)” and so on; rather we say “as-salaam ‘ala Rasool-Allah (lit. Peace be upon the Messenger of Allah)” or “Allahumma salli wa sallim ‘ala Muhammad (O Allah, send blessings and peace upon Muhammad)”, as we say in the tashahhud during the prayer: “As-salaamu ‘alayka ayyuha an-nabiyyu wa rahmat-Allahi wa barakaatuhu (Peace be upon you, O Prophet, and the Mercy of Allah and His Blessings).”
Moreover, only saying salaam upon the Messenger of Allah (blessings and peace of Allah be upon him) is also subject to further discussion. Allah, may He be exalted, has instructed us to send blessings and peace (salaam) upon him, as He says (interpretation of the meaning):
“Allah sends His Salaah (Graces, Honours, Blessings, Mercy) on the Prophet (Muhammad), and also His angels (ask Allah to bless and forgive him). O you who believe! send your Salaah on (ask Allah to bless) him (Muhammad), and (you should) greet (salute) him with the Islamic way of greeting (salutation, i.e. As‑Salaamu ‘Alaykum)”
Hence an-Nawawi (may Allah have mercy on him) said:
When a person sends blessings upon the Prophet (blessings and peace of Allah be upon him), let him combine the salaah (sending blessings) and the tasleem (sending salaams), and not limit it to only one of them. So he should not say “Sall Allahu ‘alayhi (May Allah send blessings upon him)” only, or “ ‘alayhi as-salaam (upon whom be peace)” only. End quote from al-Adhkaar (p. 107).
And Allah knows best.