Definition: Christ’s death actually paid for the sins of those whom He knew would ultimately be saved (i.e. the elect)
"Paul makes grace common to all men, not because it in fact extends to all, but because it is offered to all. Although Christ suffered for the sins of the world, and is offered by the goodness of God without distinction to all men, yet not all receive him."
"This clause is of a similar import with the former; for, as there is one God, the Creator and Father of all, so he says that there is but one Mediator, through whom we have access to the Father; and that this Mediator was given, not only to one nation, or to a small number of persons of some particular rank, but to all; because the fruit of the sacrifice, by which he made atonement for sins, extends to all. More especially because a large portion of the world was at that time alienated from God, he expressly mentions the Mediator, through whom they that were afar off now approach. The universal term all must always be referred to classes: of men, and not to persons; as if he had said, that not only Jews, but Gentiles also, not only persons of humble rank, but princes also, were redeemed by the death of Christ. Since, therefore, he wishes the benefit of his death to be common to all, an insult is offered to him by those who, by their opinion, shut out any person from the hope of salvation."
"Here a question may be raised, how have the sins of the whole world been expiated? I pass by the dotages of the fanatics, who under this pretense extend salvation to all the reprobate, and therefore to Satan himself. Such a monstrous thing deserves no refutation. They who seek to avoid this absurdity, have said that Christ  suffered sufficiently for the whole world, but efficiently only for the elect. This solution has commonly prevailed in the schools. Though then I allow that what has been said is true, yet I deny that it is suitable to this passage; for the design of John was no other than to make this benefit common to the whole Church. Then under the word all or whole, he does not include the reprobate, but designates those who should believe as well as those who were then scattered through various parts of the world. For then is really made evident, as it is meet, the grace of Christ, when it is declared to be the only true salvation of the world."
So did Calvin believe in the doctrine of limited atonement? Based on his remarks of I John 2, I would say "YES". Yet if all I had was his comments on Romans 5:18, I would probably say "NO".
My answer is this: In my opinion, Calvin was not clear on this issue, therefore, I will ask him someday in Heaven.
You believe in universalism.
You believe in unlimited atonement.
Definition: When Christ was judged on the Cross for sins, He paid for the sins of the entire human race—not just for the elect.
You are concluding that the Father poured His wrath on the Son for the sins of non-elect, who are the individuals that are going to willfully reject Him as Savior.
You are concluding that the non-elect are judged only for the sin of unbelief, not all of the rest of their sins such as murder, rape and pedophilia (since those sins were paid for by Christ on the cross).
1 John 2:2 He is the propitiation for our sins, and not for ours only but also for the sins of the whole world.
“As far as I can see, a text such as 1 John 2:2 states something about the potential breadth of the Atonement. As I understand the historical context, the proto-gnostic opponents John was facing though of themselves as an ontological elite who enjoyed the inside track with God because of the special insight they had received. [Footnote 2: I have defended this as the background, at some length, in my forthcoming commentary on the Johannine Epistles in the New International Greek Testament Commentary (NIGTC).] But when Jesus Christ died, John rejoins, it was not for the sake of, say, the Jews only or, now, of some group, gnostic or otherwise, that sets itself up as intrinsically superior. Far from it. It was not for our sins only, but also for the sins of the whole world. The context, then, understands this to mean something like “potentially for all without distinction” rather than “effectively for all without exception” – for in the latter case all without exception must surely be saved, and John does not suppose that that will take place.”
Deuteronomy 29:29 "The secret things belong to the LORD our God, but the things that are revealed belong to us and to our children forever, that we may do all the words of this law.
The primary way that God loves the non-elect is by bestowing His common grace (i.e. government, conscience and the withholding of divine judgment) on those who will reject Him.
Furthermore, I would also argue that God loves the non-elect through the universal reconciliation initiated by Christ's death on the cross (Col. 1:20).
In other words, regenerated Christians function as lights in the world, which gives "pockets of kingdom comfort" that is often enjoyed by unbelievers (i.e. the impact of William Wilberforce and the abolition of slavery).