This has to be dealt with:
"And this translation, since the promulgation of the Gospel, cannot be effected, without the laver of regeneration, or the desire thereof, as it is written; unless a man be born again of water and the Holy Ghost, he cannot enter into the Kingdom of God. (Trent, Decree on Justification, cap. iv)"
Let's simplify this with a quick substitution.
Gospel = cup of tea
Laver of Regenaration = Water
Desire thereof=teabag.
And this translation, since the promulgation of the cup of tea, cannot be effected, without the Water, or the teabag, as it is written; unless a man have tea of water and the teabag, he cannot enter into the Kingdom of God. (Trent, Decree on Justification, cap. iv)
So, the question becomes more apparent, Can you have a cup of tea by desire? or do you just have hot water?
The next big question that comes around is How is baptism of desire not "saved by faith alone?"
For, although no one can be just, but he to whom the merits of the Passion of our Lord Jesus Christ are communicated, yet is this done in the said justification of the impious, when by the merit of that same most holy Passion, the charity of God is poured forth, by the Holy Spirit, in the hearts of those that are justified, and is inherent therein: whence, man, through Jesus Christ, in whom he is ingrafted, receives, in the said justification, together with the remission of sins, all these (gifts) infused at once, faith, hope, and charity. For faith, unless hope and charity be added thereto, neither unites man perfectly with Christ, nor makes him a living member of His body. For which reason it is most truly said, that Faith without works is dead and profitless; and, In Christ Jesus neither circumcision, availeth anything, nor uncircumcision, but faith which worketh by charity. This faith, Catechumen's beg of the Church-agreeably to a tradition of the apostles-previously to the sacrament of Baptism; when they beg for the faith which bestows life everlasting, which, without hope and charity, faith cannot bestow: whence also do they immediately hear that word of Christ; If thou wilt enter into life, keep the commandments.
Let's simplify this with a quick substitution.
Your reading makes the "or" nonsense. It should really be "and", if you were right.
This translation cannot be effected without Baptism or its desire.
This translation cannot be effected without Baptism and its desire.
Equivalent? I think not.