Is glycolysis the beginning part of fermentation, or does fermentation follow glycolysis?
As @bpedit indicates in his comment, this is a semantic question — i.e. one regarding the meaning and usage of words. I will explain how I and others use these words and why. If you are convinced by my logic you will wish to use them in the same way, if not you are free to use them differently. However there is no ‘universal truth’ here, and if you are student I have no idea what your instructor thinks.

The problem is that ‘fermentation’ and ‘ferment’ (both as a noun and a verb) are old words, predating any understanding of the processes involved. Thus the Oxford English Dictionary quotes a relevant early example of the use of ‘ferment’ as a verb as follows:


  1663 Cowley Verses, to Royal Society iv, All their juyce did .. Ferment into a .. refreshing Wine.


From this, it is clear that the idea of the fermentation process is the conversion of some initial compound (the sugar in the ‘juyce’) to some final compound (the alcohol in the wine). This usage is still current in referring to the process of fermentation, for example a Google search brings up a BBC Science educational page which states:


  Beer and wine are alcoholic drinks made by fermentation reactions that use yeast to convert sugars into ethanol. 


You may or may not be aware that term ‘ferment’ was historically used to mean what we now term yeast, and ‘Zwischenferment’, the obsolete German term for certain enzymes in yeast, is derived from this. So the original concept was merely of some biological conversion effected by yeast, although this has later been extended to bacteria and the idea changed to indicate metabolic processes capable of generating ATP anaerobically.

Part of the discrepancy in definition may be the emphasis on the end-product of the fermentation — ethanol fermentation, lactic acid fermentation etc. However, it seems clear to me that the fermentation is the whole chain of metabolic events from sugar (or whatever) to alcohol (or whatever); and if glycolysis is part of that process (which it is in this example) then glycolysis is part of the fermentation. (Of course, glycolysis can occur in other circumstances where the product, pyruvate, is oxidised in the tricarboxylic acid cycle, which is not a fermentation process.)

Footnote

The extent to which one can tie oneself in knots with trying to define fermentation is illustrated in the best-selling and well-respected text book, originally written by Lubert Streyer and subsequently maintained by Berg et al. I mention this text both because an older edition is available on-line, and because a book of such value to the publishers is carefully vetted by referees. In the chapter on Glycolysis there is an attempt to define Fermentation:


  Fermentation: An ATP-generating process in which organic compounds act as both donors and acceptors of electrons. Fermentation can take place in the absence of O2. Discovered by Louis Pasteur, who described fermentation as “la vie sans l’air” (“life without air”).


So the biochemical lawyers have produced a definition that very few readers will be able to take in at first sight. What is this business about electron donors and acceptors? Well what it means in relation to the fermentation process in which lactic acid is produced (note my legalistic choice of words) is that one organic compound is reduced (glyceraldehyde 3-phosphate) — by NAD+ — and one organic compound is oxidized (pyruvate) — by NADH. And as the production of ATP is included in the definition this means that Berg et al. include glycolysis in this definition of fermentation.

…except that on the same page there is the following statement:


  pyruvate is converted, or fermented, into lactic acid in lactic acid fermentation or into ethanol in alcoholic fermentation 


So here it seems that the word is being used for the conversion of pyruvate to lactate or ethanol, i.e. it excludes glycolysis.

Pasteur managed to talk about fermentation without being aware of glycolysis or ATP, and it is clear to me that you can write whatever carefully phrased definitions you like, but people are going to continue to use venerable terms like fermentation in whatever way seems natual to them.