Advice
Talk to your advisor. At the end of the day, your goal is to complete a degree. The primary obstacle[1] to completion is your advisor, followed by your committee. If your advisor has a strong preference, do what they tell you to do.[2]
Read other theses in your area. Particularly from your institution. See what other people in your department have done in the past, and what other people writing on similar topics have done in the past. Your institutional library should have a collection of past theses by people at the institution—have a look at what other people from your department (and even advisor) have done, and emulate their structure.
Refer to your institution's guides. Your institution will have a guide for theses. It will (perhaps) tell you about which fonts you can and cannot use, how wide your margins should be, how sections should be numbered, what pages are necessary, and so on. It is possible[3] that this guide has some guidance about how to organize your thesis. If it does,[4] do what it tells you to do.
Read texts in your area. See what other people in your field are doing. If appendices of notation are common, go that way. If introductory sections of notation are common, do that.
Do what you like. Ultimately, at the end of the day, your thesis is your document. You get to decide about style. You have expressed a dislike for appendices, so don't include one! The reality is that none of the above points are likely to apply. Most advisors aren't going to care about the structure of your thesis, your institution's style guide is likely silent on the issue, and I would imagine that there is no consensus in your field.
Anecdote
For what it is worth, my own solution was to include an index of new, idiosyncratic, or otherwise non-standard[5] notation just after the table of contents and the list of figures. I then introduced notation in the basic flow of the text, and had LaTeX build the index of notation automatically. A lot of notation was introduced in my first two chapters (background and introduction), but I continued to introduce notation throughout the document. I think that this worked pretty well.
Notes
[1] My use of "obstacle" here is a bit tongue-in-cheek. Hopefully, your relationship with your advisor is good, and they are an active help, rather than hinderance.
[2] For example, my masters advisor very strongly disliked the Oxford comma. I very strongly disagreed with him. There are no Oxford commas in my masters thesis.
[3] Possible, but I think highly unlikely.
[4] And it probably doesn't.
[5] There is absolutely no need to define standard notation in your field. Don't waste your readers' time with that.