I am confused by your post. Please don't ask why I am confused. I don't know. However, I may have provided you with something you are confused by. The second resurrection is not spoken of. What I am saying is that not all end up in the lake of fire unless it says so. So, what does it say?
1.
You are right in pointing out that the term "the second resurrection" is inferred rather than specifically named. What I meant was when the "rest of the dead" live again (Revelation 20:5 & Revelation 20:12-13.)
2,.
You are correct in that it does not state that all (of the rest of the dead) end up in the lake of fire. It actually says:
Revelation 20:14-15 KJV
(14) And death and hell were cast into the lake of fire. This is the second death.
(15) And whosoever was not found written in the book of life was cast into the lake of fire.
coupled with
Revelation 21:6-8 KJV
(6) And he said unto me, It is done. I am Alpha and Omega, the beginning and the end. I will give unto him that is athirst of the fountain of the water of life freely.
(7) He that overcometh shall inherit all things; and I will be his God, and he shall be my son.
(8) But the fearful, and unbelieving, and the abominable, and murderers, and whoremongers, and sorcerers, and idolaters, and all liars, shall have their part in the lake which burneth with fire and brimstone: which is the second death.
Summarizing:
1. The resurrection of the rest of the dead is promised (even of those who are not righteous) to which Paul confirms his belief in Acts.
2. Although there are those in Christ who are already raised (this is called the first resurrection) those who were dead but were had not already raised will be alive for that judgment
3. On the other side of that judgment, there are two possible results:
a) an ending, to be destroyed in fire, also called the second death. The opposite of life.
b) a beginning, to enter into life and the holy city, to drink of the waters of life that flow freely for all that thirst. Eternal life, the opposite of death.