Yes, you want more images if you're going to use more vectors, but you do not necessarily need more vectors if a low number of vectors gets an accurate enough image. I'm personally not seeing what you think is inaccurate about these images. I think they're pretty spot on.

The problem with more vectors is that while you can get more precise, you can also lose the ability to place the subject in different scenarios, as it will start memorizing details from the locations, outfits, photo styles, etc. Early attempts with a high number of vectors had these kind of issues for me.

I tried with 1 vector and it just wasn't realistic enough to my eye (although SD 1.5 did better with 1 vector imo). 2 vectors looks pretty spot on to my eyes, and I'm a massive fan of Emma. There are little things that can be improved on, and not every render looks perfect, but overall, it looks like a good representation of her likeness to my eyes. So I could certainly do a third vector to see if there was any noticeable improvement, but I stopped here because I thought it looked good as it was.

The point of an embedding is to completely replace how the model interprets the words you type, so it SHOULD be used every time you type those words. If you want a different functionality, feel free to rename it yourself. The only time I'd name an embedding to something other than the natural language I'd use to describe that thing, is if I have multiple embeddings of the same subject.

The actual name of the embedding doesn't change how training works. With [filewords] every photo has been named manually by myself, and every photo contains the name Emma Stone in it. When training, it replaces the text "emma stone" with the embedding, and uses the rest of the text to build the prompt around it, and that additional context around "Emma Stone" is useful for it to be able to find the ideal vectors to replace those words with, so that it doesn't start grabbing details from her outfits, or her locations, etc, but focuses on her, because the other words already describe those other details in the image.

So whether you call it Emma Stone or EmmaStoneEmbed or whatever, you'd have to name all of the images to match that in order for the TI process to isolate the correct words, and I don't know about you, but it's just easier to use the real name when naming my images, especially if I plan to use that same dataset in a Dreambooth training for example. So it's best to train based in the natural words, and then if you want to rename the embedding file, you can do so yourself.

An embedding will always be based on one specific model, and work best with that model, but any model that has been trained on the same base will usually at least be able to make use of the embedding and it should still have some similarity. How similar it will look will depend on what weights the embedding activate in the neural network, and how much those weights have or have not been changed during finetuning.

I have actually tried to train embeddings on some of the models I've downloaded from here, but at least on A1111, the training seems to produce completely random results, so A1111 at least seems to be tied to the base models when training embeddings. You can however train a hypernetwork on any model though.