And what about the interrobang‽
I don’t mean to sound snippy either. But you have nothing to back up her statements; all the knowledgeable people here are telling you that you and she are wrong; and the cites I’ve found reinforce that. Any objective judge would rule against you.
I’d have to know how they handle nested quotes. But the main issue for me would be that I can’t type them easily on my keyboard.
There are also the single-quote versions of those: ‹ and ›
Here is a page listing a whole bunch of different kinds of quote marks including their Unicodes, their html codes, and examples of their usage in two dozen different countries.
I wouldn’t be a huge fan of single quote version, as they look to much to me like < and >.
I imagine one could always tell which is meant by the context. Similarly, the dot we use to end a sentence looks confusingly similar to the dot we use in writing decimal fractions (as used in the United States anyway), and we can usually distinguish those from the context.
My father used to muse on the feminine demonym for a person from Michigan.
Yeah, few people even recognize Michigoose unfortunately 
What, it isn’t Michigonian?
One of the changes in the US edition of Harry Potter was changing “revise” to “study”. (Incidentally, I refuse to put that period before the quote mark since the quote was not a sentence.)
But one thing that irritated me enormously when a British copy editor wanted to delete subjunctives in a book I was a coauthor of. In “A necessary and sufficient condition that the category \cal C have” and she wanted to change “have” to “has”, dozens of times. We insisted and she left it. In the end, they couldn’t deal with the LaTeX and we mailed them camera-ready copy which they printed.
I saw that period after the quote 
Interesting about the editor. Some editors are tyrants. I’m glad you stuck to your guns
I tried to work with LaTeX years ago and it was just too hard for me. The owners at the company though it would be great if I learned it and we could offer it as an option. Never mind I had a half dozen other things on my plate. Thankfully I just learned the basics and the idea never went further. But they keep piling more and more stuff on me. The place will be the death of me
That’s great that you could provide the camera-ready copy. 
Anyone else shaking their head at the OP assuming that everyone here has noticed their posting style, and kept track of which posts were made by him, and mentally cataloging the punctuation? And then the OP is assuming… what? That we’re all fuming at his single quote marks? Or mentally giving him a high-five for saving electrons with his single quotes?
Sigh, I’ve run into this here so many times (usually by a poster that’s full of themselves and is starting most of their posts with “I know everyone here hates me for ______ …”), that I really want to start a thread titled “Get over yourself…”
If the OP is not one of those posters, I apologize … which proves my point that people here aren’t keeping track of everything a specific person has written.
Although I am American, I edited using British conventions for about eight years and thought I was quite familiar with British v. American English spelling, vocabulary, and punctuation.
But I did once find myself in a ridiculous fight on line, regarding costs at international schools serving expatriates. Everyone was adamant that they were correct.
It turns out that the cause of the disagreement was different understandings of the meaning of “tuition.” I and the other Americans used “tuition” to mean the amount of money it cost to attend the school. The British participants were using “tuition” to mean what in America we would call “tutoring.”
When both sides finally realized the problem, we were able to resolve the argument easily.
I was surprised that I hadn’t known of that difference between American and British English. I guess it was because the editing work I did was in the field of infrastructure finance. Education issues never arose.
Michigyne.
What you/the editors can do is provide the authors with a LaTeX “class file” or template implementing your house style. But then you have to go through the final submission anyway and fix all the problems, including quotation mark and punctuation issues. And it is only relevant if the publisher uses LaTeX internally as part of their process; some tell you flat out that they prefer to receive Microsoft Word files.
But, back to quotation marks,
I make no claim as to the relative frequency of different conventions, but I can disprove the statement “British publications always start with single quotation marks on the outside” very easily:
(London, 1889.)
ETA I have not heard of triple, quadruple, etc. quotation marks either, nor do they seem to be in the Unicode table mentioned above.