Interesting to also note that Wesley considered his view within a "hair's breadth" of Calvinism regarding the Depravity of Man.I'm not sure I understand your question but the notion that "regeneration precedes faith" doesn't work well with most of the NT and many Calvinists modify this to where regeneration precedes faith precedes justification. Justification may or may not include some sort of salvation experience. The problem is that many Calvinists have never had any kind of salvation experience so it doesn't matter to them where the very non-tangible concept of regeneration is placed in that sense. Their theology forces it to precede faith however. Some Calvinists go as far as separating out the new birth from regeneration to where it becomes regeneration precedes faith precedes the new birth but I don't think this is technically within the scope of the theology. In the end, regeneration may refer to nothing more than the notion that God has to start the process of salvation, the sinner cannot.
What many Calvinists battle, is the notion that the sinner simply decides to believe of his own accord and with no influence from God and then he is regenerated. That is how they commonly interpret the opposite "faith precedes regeneration". This notion is commonly attributed to Charles Finney, which is probably not entirely correct. John Wesley made this a bit clearer than James Arminius did, to where it became "prevenient grace precedes faith precedes regeneration". Prevenient grace may amount to a very tangible conviction and drawing of the Spirit but it may also be very general. It can be based on the sinner being exposed to the gospel or elements of it like Catholics are or it can be as general as the Spirit drawing on people who haven't heard the gospel at all, to at least seek out the real God (meaning that will lead them to encounter the gospel later). The main difference is that the Calvinist concept of regeneration as the first step in the process cannot be resisted and rejected, while prevenient grace certainly can. Besides that, the resulting process may look largely similar. Something has to happen first then faith may come about and only then the person is seen as justified by God.