I consider myself a classic moderate Calvinist. The term 'moderate' itself with respect to Calvinism goes back to the 1600s. These Calvinists held to an unlimited expiation in the sacrifice of Christ.
This is not 4-point Calvinism, nor is it necessarily Amyraldism. Though highs very often throw these accusations at us moderates, and even the insult 'Arminian,' when frustrated. Classic moderate Calvinism believes in a kind of limited atonement, proper limited atonement - an unlimited expiation/limited intent variety. This view was represented by many delegates at the Synod at Dort, and the Canons of Dort were written broad enough to allow moderate Calvinists to sign. Some of the Westminster Assembly Divines, like Calamy, held to this view also.
The classic moderate view is also that of the original Reformers, including Calvin (see the Heidelberg Catechism Q & A 37, together with Ursinus' commentary on this Q & A). Classic moderate Calvinism best explains all scripture teaching on the atonement, both general texts and texts that teach a special intent. It embraces real sufficiency as it was meant in the Lombardian formula - sufficient for all, efficient for the elect. This view is consistent with the sincere offer of the gospel to all.
High-Calvinism is a Calvinism that goes higher than the original Reformers and Calvin. This occured with Beza's supralapsarianism, and with a limited expiation view of the atonement. Hyper-Calvinism is the highest form of high-Calvinism. That is almost axiomatic. High-Calvinists often put it in either/or fashion this way: either Christ died to make certain the salvation of his people; or his death merely made the salvation of all possible. Classic moderate Calvinism presents it as both making the salvation of the elect certain, and making it possible for all to be saved.
Hyper-Calvinism manifested itself with a denial of the Free Offer of the gospel and other things like Duty Faith, denial of the love of God for all including the non-elect, the denial of common grace, a denial of a preceptive will of God, etc. Hyper-Calvinism is one, or more, of these beliefs. For the hyper-Calvinist there is an over-emphasis on sovereignty and the will of decree, which eclipses human responsibilty.
I think hyper-Calvinism is the logical result of the limited expiation of high-Calvinism. I would argue that the free offer and duty faith are not consistent with the limited expiation of high-Calvinism. For if the expiation is strictly limited to the sins of the elect: 1) how can all men be genuinely offered salvation; & 2) how could all be commanded to obey the gospel? Nor can the work of Christ be sufficient for the forgiveness of the sins of all in this view. Sufficiency for high-Calvinists becomes a hypothetical sufficiency - i.e. if more had been elected, the value is such that it would be sufficient, etc.