Harmonic Chant is another popular name for Overtone Singing. By using the word ‘Chant’ it appeals to a slightly different audience associating it to Gregorian Chant and giving it a more spiritual context. However, this is mostly just marketing. Personally I see Overtone Singing as the generic overarching term for all such vocalisations that seperate the Fundamental from the Overtones audibly. Harmonic Chant is a subset preferred by certain artists such as David Hykes.

There is a technical difference as well which is that all harmonics are Overtones but not all overtones are harmonics. Harmonics are by definition  perfect whole number ratios of the fundamental frequency. Overtones can be distorted by physical anomalies such as tightness and thickness of a string.

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The Partial Series is another name for the overtone series except that technically the partial series considers the fundamental as the first partial whereas the overtone series considers the fundamental separate and starts counting from Partial number 2 as the first overtone. This creates much confusion because many practitioners using the numbers to name the overtones will actually use number 1 as the fundamental but call it the overtone series instead of the partial series.

The reason is one of convenience as the powers of 2(2 squared, 2 cubed, etc) always elicit an octave above the previous power of 2. This means that a number 8 (Technically the 7th overtone) is an octave (More specifically the 3rd octave above the fundamental).

I also use number 1 as the Fundamental and encourage everyone to do the same. All the maths fit together when adding and multiplying ratios.

Example a perfect 4th (4/3 ratio) plus a perfect 5th (3/2 Ratio) is an Octave (4/2 after cancelling the 3s).