Literally "Islam" means "submission" so "Islam" is called "Islam" because the sole criteria which has to be satisfied in Islam is one's submission to the will of God.
Literally speaking, all Divine religions from Adam to Mohammad can be called "Islam" because they share the same criteria as mentioned before. They all require man to surrender himself to only One God and worship none other than Him.
On the other hand as we all know, Islam is the last religion whose book is Holey Quran which was brought down by Prophet Mohammad(pbuh).
The word "Islam" has been used in Quran in both ways:
إِنَّ الدِّينَ عِنْدَ اللَّهِ الْإِسْلامُ وَ مَا اخْتَلَفَ الَّذينَ أُوتُوا الْكِتابَ إِلاَّ مِنْ بَعْدِ ما جاءَهُمُ الْعِلْمُ بَغْياً
بَيْنَهُمْ وَ مَنْ يَكْفُرْ بِآياتِ اللَّهِ فَإِنَّ اللَّهَ سَريعُ
Indeed, with Allah religion is Islam, and those who were given the Book did not differ except after knowledge had come to them, out of
envy among themselves. And whoever defies Allah's signs should know
that Allah is swift at reckoning.(3:19)
In this verse, the word "Islam" has been used in the literal meaning because of the rest of the verse. The first part of the verse is followed by some sentences describing how the People of Book differed among themselves. In order for the first part to make sense we have to say that the literal meaning of "Islam" is intended in the verse. So the meaning of the verse can be explained as follows: "The religion with Allah is only one which is submission to the will of Allah and accepting willingly His commands and ordinances. Although these commands and ordinances differed in quantity and quality from one religion to another due to the various levels of perfection in every religion, they are all the same in essence owing to the fact that they were all conveying the same message which was surrendering oneself to the will of God. In spite of knowing the truth and being aware of the fact that the religion with Allah was only one as explained before, the People of Book differed among themselves on account of envious and jealousy.
The word "Islam" has been used in the second meaning as well:
وَ مَنْ يَبْتَغِ غَيْرَ الْإِسْلامِ ديناً فَلَنْ يُقْبَلَ مِنْهُ وَ هُوَ فِي الْآخِرَةِ مِنَ الْخاسِرينَ
Should anyone follow a religion other than Islam, it shall never be
accepted from him, and he will be among the losers in the Hereafter
It should be noted that Quran can be interpreted by itself which means that verses interpret each other. The above-mentioned verse is clearly saying that your understanding of the verse 62 of Surah Baqarah is not correct. There is only one religion accepted by Allah. AS a result the verse 62 of Surah Baqarah is interpreted as follows:
No one go to the Paradise because they are called Muslim, Jew or Christian. Whoever believes in Allah and the Last Day and act righteously will be rewarded by Allah and now that the Quran has been sent down, whoever believes in Allah consequently should believe in His last Book, Quran which will result in believing in the verse 85 of Surah Al-e-Imran. The same rule applies for the people who lived at the time of revealing other Holey Books.
Moreover, if we accepted your interpretation of the verse, how would you justify numerous Quranic verses inviting People of Book to convert to Islam?