For what it's worth, my entire point is that there is no line. A line is completely unnecessary. Just like a native English speaker doesn't need to be told what the rule is for adjective ordering -- "the big red dog" sounds right, "the red big dog" sounds wrong.
I've had perfectly fine veggie sandwiches, when I've ordered vegetarian sandwiches, but that wasn't my question -- would you, personally, consider that to be a reasonable response if you went into a random restaurant, or if you were asked to prepare a sandwich for a customer who said they didn't care what kind of sandwich?
I'm not asking rhetorically -- I'm genuinely curious. People do differ on this; many people I talk to disagree on the bagel question, and consider both variants a sandwich or consider neither one a sandwich. My theory here is that I grew up eating bagels at home and not at restaurants, so would eat my bagels with cream cheese open faced, and not sandwiched between the two halves, so I never really created the association with being a sandwich; while people who consider it a sandwich grew up in cities where you got a bagel from a store or deli, where it is pretty much universally served with the two halves stacked, and thus is more sandwich-like during developmental years.
Would you feel comfortable, without knowing the customer, giving them such a minimalist veggie sandwich and still expect that they would be satisfied that you've obeyed the broad parameters?