This is the house that I lived two years ago.This is the hou

This is the house that I lived two years ago.This is the house I lived two years ago.
这两个定语从句对吗?为什么?
某些在从句中充当时间,地点或原因状语的"介词+关系词"结构可以同关系副词when 和where 互换,例如：
This is the house in which I lived two years ago.
=This is the house where I lived two years ago.
That有时可以代替关系副词 when,where 或者why引导定语从句表示时间．地点或原因（百度百科）
那么,是否上面一句句子可以改为：
This is the house that I lived two years ago.
That有时可以代替关系副词 when,where 或者why引导定语从句表示时间．地点或原因,that也可以省去.那么,是否还可以演变成：
This is the house I lived two years ago.
如果错的话,错在哪儿了?