Word – Microsoft Equation 3.0 with Word 2013

Well, you sort of answered your own question, unfortunately. The newer Equation editor is not 100% compatible; some characters will turn out different in older versions of Equation editor. I think there is a way to save the formulas in Word 2007 using the compatible version, although I can't remember if it's possible.

You can read more about it in this article. Here's a short summary:

An equation created using the new Word 2007 equation editor will appear as an image that cannot be edited when you subsequently open the document in Word 2003. This is the case whether you saved the document in .docx (Word 2007) or in .doc (Word 2003) format.

However, it also shows how to create an equation in Word 2007 that can be read and edited in Wrod 2003.

These are the same format. My company, Design Science, makes the Equation Editor that comes with Microsoft Office on Mac and Windows since 1991 or so. MathType is the fancy version of Equation Editor with lots more features, including the TeX input feature you mention. Your publisher should be encouraged to mention that it accepts both MathType and Equation Editor equations as most publishers do. I suspect they have been getting documents containing MathType equations and handling them just fine all along.

By the way, the next version of MathType will feature compatibility with Mac Office 2011 which is due out later this month. As you may know, Microsoft left out Visual Basic for Applications (VBA) support in Office 2008 but they are bringing it back in this new version. That means MathType will have more commands like it has in Office 2004. In particular, you will be able to insert new equations without going through the Insert Object process.