Last time/The last time
When you are using "the last time" to mean "the previous time something happened", then the use of "the" is optional. All of the examples you quote are correct and mean the same without 'the".
With other meanings of "the last time", especially "the final time" you cannot omit "the".
CORRECT: This is the last time I am going to do it.
INCORRECT: This is last time I am going to do it.