I rescued
the below from the Wayback machine. But it doesn't address the restaurant situation explicitly.
3人と3名、その違いは？
人数を数える時に使う「人（にん）」と「名（めい）」、使い分けはどうしたらいい？という質問をいただきました。放送においては、話し言葉に近いという理由から「人」を使うことが多いのですが、日常的には何となく使い分けてもいます。（1）「今年の新成人は、およそ127万□」（2）「250□の新入生を迎えました」（3）「△△様はじめ3□の方が出席されます」どうでしょうか？（１）は全体数をざくっと数える感じ。「人」を使うことが多いですね。（2）は「人」でもよさそうですが、入学式など改まった場面では「250名」と言い換えることもあります。（3）は△△様と相手を敬った丁寧な表現。こうした時には「名」の方がしっくりきませんか？このことから、「人」＝【単純に人数を数える場合】、「名」＝【改まった表現をする場合、定員･定数のある場合】とまとめることができそうです。中央大学の飯田朝子教授は、"参加者数、定員数、合格者数、卒業者数など、リストとして名前を書き出すことのできる人数は「名」で数える"とも定義しています。一人ひとりが特定できる場合は「名」がふさわしいというわけですね。より使い分けがはっきりしているのは、他の言葉と結びついた時。例えば、「３□目」「２□様」といった表現では「3人目」「2名様」と決まった言い方しかできません。「お1人様」「お2人様」という言い方もしますが、これはちょっと特殊です。他にも、出前などは「4人前」、車は「5人乗り」など、慣用的にどちらかが使われている例もありそうです。結論としては、｢人｣と「名」どちらを使っても何ら問題はありません。
その場にふさわしいと思う方を、自分の感覚で選んで使えばよさそうです。
[(mostly) auto translation:
How should I use "nin" and "mei" when counting the number of people? I was asked this question. In broadcasting, we often use ｢人｣ because it is closer to the spoken language, but we also use them interchangeably in our daily lives.
(1) "The number of newcomers this year is approximately 1,270,000□,"
(2) "We welcomed 250□ newcomers,"
(3) "Mr. △△ and 3□ other people will attend the meeting,"
What do you think?
(1) is a rough count of the total number and ｢人｣ is most commonly used.
(2) ｢人｣ may also be used but on formal occasions such as an entrance ceremony, it may be rephrased as "250名".
(3) △△-sama is a polite expression of respect for the other person. In such situations, "名" would be more appropriate, don't you think?
From this, it seems that we can summarize ｢人｣ = "when simply counting the number of people" and 「名」 = "when using a formal expression or when there is a fixed capacity or fixed number of people."
Professor Asako Iida of Chuo University also gives the definition of「名」as the number of people whose names can be written down as a list, such as participants, quota, successful applicants, and graduates. So, if each person can be identified, 「名」 is appropriate. The distinction between the two is clearer when they are used in conjunction with other words. For example, "3□me" and "2□sama" can only be used to refer to "3人" and "2名様," respectively. The phrases "one person" (お一人様) and "two people" (お二人様)are also used, but these are a bit more specific. There may be other cases where one or the other is customarily used, such as "4人前" for delivery, "5人乗り" for a car, and so on. In conclusion, there is no problem in using either ｢人｣ or 「名」.
It seems that it is fine to use whichever you feel is appropriate for the occasion, based on your own sensibilities.
And this video for beginners interestingly tries to cover both even though most videos for beginners don't mention 名 at all.
