For the two phases “the last thing I want is…” and “the last thing that I wanted was”, can you give some examples of usage and explanations in what situation when you want to use one over the other?
The last thing I want is to hurt you
This means that the speaker is doing (or is going to do) something that involves a risk of hurting the listener, but is stating explicitly that that is not something he wants to do—quite the contrary, he is very reluctant to hurt the listener. Crucially, the listener has not yet been hurt.
The last thing I wanted was to hurt you
In this version, the past tense, the speaker knows that he has already hurt the listener. He is now trying to atone for this by saying that the listener being inadvertently hurt was not what the speaker wanted—it was an entirely unwanted side effect of whatever it is the speaker did.
In other words, there is nothing strange about these two phrases: one is in the present tense and deals with a present-future outcome, while the other is in the past tense and deals with a past (= already known/existing) outcome.