This post is probably best answered by a miaphysite since most of the questions are gear towards them. Im sure they do have their own fathers and teachers that have written treatise on these things through the ages. The OO and EO do say very close to the same thing and in actuality so do the Assyrians! But simply attributing these disagreements on semantics (as many in orthodox forums are quick to do), semantics is an oversimplification and overlooks the subtle variations.
For instance in (eastern) Orthodoxy, a person must accept both of the following formulas as orthodox; that Christ is OF two natures and that He is IN two natures. When applied correctly in context the former denounces nestorianism while the latter denounces monophysitism. In the 4th council, Dioscorus was condemned for eutychianism for only accepting the formula that Christ is of two natures but not accepting the formula that he is in two natures. In OO the latter phrase is a heresy, in the EO the rejection of such a phrase constitutes a soft monophysitism refusing to acknowledge that such a heresy exists or that nestorianism is a worse heresy making eutychianism a tolerable heresy.
OO believe in a composite nature, will & energy. Anotherwords there not comfortable with acknowledging a DISTINCTION between the 2 natures, wills and energies. Hence they label themselves as miaphyisites, mia= one nature (which can allow for a composite nature) as opposed to monophysite, mono- one in an absolute sense, 'one and only one'. As you can see there rejection for the label of monophysitism is that they are aware there are two natures, but act as one composite nature (mia) due to the uniqueness of the hypostatic union , but reject diophysite as this implys a discernable distinction.
From an Oriental Orthodox Perspective Eutychiansim is just as terrible a heresy as Nestorianism and Appolinariansim.
We acknowledge that Christ is completely God and Completely man, lacking nothing in either nature.
St. Cyril stated that we distinguish the two natures (two physis) is Christ "in throught alone" when speaking about the two natures.