For relative time in talking/writing:
"前一天", "同一天/当天/第一天", "后一天", "第三天", "第四天", etc.
(supposed to be.)
For current time:
"前天", "昨天", "今天", "明天", "后天"
-2 day, -1 day, 0 day, +1 day, +2 day
今天可以是上班第一天，明天也可以是休假第一天, "第一天" is a relative time, it just ...
The quoted dialogue is mostly correct. Unlike 明天 (tomorrow), 第二天 (the next day) does not assume when the speaker is at. If 明天 is the day after today, 第二天 can be the day after any date. Therefore, one says sentence 1.1 but not 1.2:
A more literary equivalent of 明天 and 第二天 is 翌日:
(1.3) 門市翌日開張。(=明天 if this is found on ...
You have to first understand what "第" means. It's used to point out the position of something from a sequence.
One, two, three, four, five, six: 一, 二, 三, 四, 五
First, second, third, fourth, fifth, sixth: 第一, 第二, 第三, 第四, 第五
So 第二天 isn't completely equal to "next day", but "the second day".
当天: same day (this is correct)
Why is 来 used?
It's a style thing (and an indication of an older language, I think).
You may choose not to use it:
Also, you may write:
来 and 去 are both used in this way as directional indicators, just like in German. (English has largely given up this practice.)
Language makes a lot of shortcuts: if you use something, you ...
The answer to the question explained the function of 来 as a preposition (in order to; so that) that connect two related verb phrases, but here '用 ~ 来' can be simply explained as 'use ~ to'; (被)用来 as (be)used to
"(把)'东西'用来 表示人" = "(take) 'things' to be used to indicate human"
"用('东西')来 表示人" = "use ('things') to indicate ...