Calvin writes:

The judgment which ought to be formed concerning the visible Church which comes under our observation, must, I think, be sufficiently clear from what has been said. I have observed that the Scriptures speak of the Church in two ways. Sometimes when they speak of the Church they mean the Church as it really is before God—the Church into which none are admitted but those who by the gift of adoption are sons of God, and by the sanctification of the Spirit true members of Christ. In this case it not only comprehends the saints who dwell on the earth, but all the elect who have existed from the beginning of the world. Often, too, by the name of Church is designated the whole body of mankind scattered throughout the world, who profess to worship one God and Christ, who by baptism are initiated into the faith; by partaking of the Lord’s Supper profess unity in true doctrine and charity, agree in holding the word of the Lord, and observe the ministry which Christ has appointed for the preaching of it. In this Church there is a very large mixture of hypocrites, who have nothing of Christ but the name and outward appearance: of ambitious, avaricious, envious, evil-speaking men, some also of impurer lives, who are tolerated for a time, either because their guilt cannot be legally established, or because due strictness of discipline is not always observed. Hence, as it is necessary to believe the invisible Church, which is manifest to the eye of God only, so we are also enjoined to regard this Church which is so called with reference to man, and to cultivate its communion.

By “elect” Calvin means those has actually “appointed”–those who have been the objects of his intervention.  Thus, the Invisible Church can actually be “entered.”  Once can be a stranger and alien to the Invisible Church and then be made a part of the family.  “Elect” means “chosen,” and when one extends a platter of cookies to a friend ans says “choose one,” the offer calls for an actual apprehension with the hand, not only a mental operation in the brain.  So Calvin can speak of the “elect” as those who have been regenerated, even though he knows that, in the sense of being elect from eternity past, God had chosen others who were not yet regenerated.
(By the way, it is important to remember that Calvin was exegetically and pastorally responsible with his use of this distinction.)

But consider the Westminster Confession:

The catholic or universal church, which is invisible, consists of the whole number of the elect, that have been, are, or shall be gathered into one, under Christ the Head thereof; and is the spouse, the body, the fullness of him that filleth all in all.

Here the Invisible Church is simply a label for the group of people who have ever been predestined for salvation and resurrection glory.  One no more “becomes” a member of the Invisible Church in the WCF sense, than one becomes predestined to salvation.

Finally, if you read Calvin’s preface to the King of France, you will find that he uses the language of the Visible Church over agains the Invisible as a way to speak of rather non-glorious underground institutions, over against politically powerful and wealthy institutions.

Of course, all this is merely a difference in chosen terminology to serve certain uses.  It doesn’t represent any real difference in opinion.