Question

that在定语从句中什么状况下可以省略

除了关系词在从句中作宾语的情况，这个我已经知道了
The reason (that) he did not attend the meeting was that he was ill 这句话为什么能省略

Answer 1

You can work out this problem (in) this way. Let me explain it (in) another way.表示行为方式的way前面的介词in美国人口语习惯上省略。所以说this way名词短语作宾语但相当于作状语而不是通过介词短语作状语。所以说引导way的关系词可以是关系副词where(实际上不用)也可以是关系代词that但不用which，或用in which当然也可以省略。例如：I don't like the way (that/in whinch）he speakes to his mother.
the reason和the way一样只不过省略for，但可以用关系副词why来引导。这都是美国人的习惯问题，没有为什么，可以在网上搜关于省略句的文章。

Answer 2

老兄，那个that 引导的不是定于从句，而是一个同位语，he did not attend the meeting 是reason的同位语。后面的that引导的是表语从句。
至于that 引导定于从句时的省略，应该是在that在从句中做宾语的时候可以省略。例如：
That is the man （that) I talked with yesterday.

Answer 3

这句句子不对吧 应该是 The reason why he didi not attend the meeting was that he was ill.

Answer 4

只有一种。就是你说的