No. the second death is the lake of fire. Rev. 20:14.
You have the first death is spiritual, the second death is the lake of fire, so where do you put physical death?
True. Ultimately that is where those who are dead in trespasses and sin end up. It is all part of the same judgement - spiritual death and death eternal in the lake of fire. As to the text, there may be a little room for interpretation here, though most literally you are correct. King James says that “death and Hell were cast into the lake of fire. This is the second death.” This makes it sound as if the punishment of death itself rather than the lake is the second death. Most other versions say “the lake of fire is the second death” or “this is the second death” the lake of fire. So, saying that the lake of fire is the second death is correct. But also saying that those who are without Christ are under the sentence of the second death is also correct, wouldn’t you say?
Here are a few more verses from Revelation about the 2nd death:
Revelation 2:11
He that hath an ear, let him hear what the Spirit saith unto the churches; He that overcometh shall not be hurt of the second death.
Revelation 20:6
Blessed and holy is he that hath part in the first resurrection: on such the second death hath no power, but they shall be priests of God and of Christ, and shall reign with him a thousand years.
Revelation 20:14
And death and hell were cast into the lake of fire. This is the second death.
Revelation 21:8
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.