Which of the following is correct or preferable?

1) That's [Liz Taylor] sporting that white mane.
2) That [Liz Taylor]'s sporting that white mane.

I vote for #1.

Number one is correct if you mean to say "That is Liz Taylor sporting that white mane.

Number two is correct if the first "that" is to be stressed. "That Liz Taylor is sporting that white mane." This usage implies a negative opinion of Liz Taylor, unless there are several people named Liz Taylor and the word "that" is used to designate which Liz Taylor is sporting the white mane. Of course, designating which person is sporting a white mane isn't really necessary, is it?

They have different meanings.

"That's Liz Taylor..." means 'I identify Liz Taylor as the person with the white mane'

" That Liz Taylor's..." means 'Liz Taylor, the real Liz Taylor,the famous person of that name, not just anyone... 'or ' Liz Taylor, who thinks herself special....' is sporting a white mane . The tone of voice of the speaker tells us whether there is an innuendo.

The grammatical difference is that in #1 'that's' is short for 'that is' and so 'that' is a pronoun. The speaker or writer could say or put 'the person who is called'or 'the woman who is' instead. In #2 'that' is a determiner to indicate a specific individual. "Taylor's " is short for 'Taylor is ' and 'that' indicates which 'Liz Taylor' is meant, just as it would if the statement was " I bought that book yesterday, not this one". Now, it is unlikely that either the speaker or hearer is thinking either of the possibility that there are two people called Liz Taylor in the room or of any misunderstanding over which Liz Taylor is referred to, and so needs to indicate that the 'Liz Taylor' is one previously mentioned However, we sometimes use the determiner 'that' as a means of emphasis and it would be so used here. The speaker may mean 'the famous Liz Taylor' or 'the one and only Liz Taylor'. Certainly in British the use of 'that'in this context usually serves simply to indicate that the subject is famous, or well- known both to the speaker and to the hearer. However, it is sometimes used to show disapproval e.g. 'that Liz Taylor who thinks herself so grand,who thinks of herself as the Liz Taylor'. Gossips are fond of this device for showing disapproval : " Her at number 42. Oooh, I can't stand that woman. Do you know what that woman said to our Billy ...?". This is said with a deliberate avoidance of 'she' or 'her' or even 'the woman'

In a recent telephone conversation, someone said, "That's the whole point!" The subject was the importance of understanding a difficult process described in the last book I wrote. My belief is "that" refers to "the importance of understanding the difficult process."

1) That's [the importance of understanding the difficult process] the whole point!
2) That [the importance of understanding the difficult process]'s the whole point!

I guess that I have to change my vote to #2, because the person was emphasizing in sync with me. Are you able to make a choice either way here with the given information? I think that you will agree. (I suppose that it would have been helpful to have recorded the conversation, but that wouldn't be too nice or legal without the person's permission!)

Frankly, all along I thought that I had to use the second construction, but, of course, I saw the first construction in an article recently, and it threw me astray. It all boils down to frequent use of a language, common sense, and a good ear. Honestly, I did not think that in different instances, one or the other construction could be valid: I thought that only one construction would be always correct. How creative and expressive our language is!