Using the present tense implies that whatever was said or written has present relevance; for example because a decision has to be made or action taken based on an interpretation of the message.
It is certainly more likely that a recent message will have present relevance, but it is also true that something written some time ago may require a present interpretation and consequent action. In such a case What does he mean by that? is perfectly acceptable.
If the past tense is used for the question, no implication is made as to the present relevance of what was said. The message may or may not have present relevance.
As an aside, it cannot be that the past tense may 'signify that his (the speaker's) meaning has changed'. Whatever the speaker meant at the time of saying it remains unchanged. What may have changed in the meantime is the speaker's position or opinion on the topic in question.
All three of your examples use a "that" which is a marker of clausal subordination: in your case, that "that" marks the beginning of a declarative content clause.
Sometimes that "that" marker is obligatory, sometimes optional, sometimes not allowed. There are some related rules on this, but there is no one simple general rule. It's something that native English speakers just pick up while speaking and listening and reading.
Note: Your "that" is NOT a relative word or relative pronoun. (There is another "that" which is the marker of clausal subordination for that-relative clauses. Some grammars consider that the two markers to be, or could be considered to be, the same. In any case, the "that" marker for relative clauses has a different set of rules in regard to its presence or absence than the one marking declarative content clauses.)
One rather firm rule--well, somewhat a firm "rule"--is that if the declarative content clause is the subject of the main clause, then the "that" marker is obligatory. That is so the reader will get a heads-up to realize that the content clause's subject is NOT the subject of the main clause, even though it is located at the beginning of the sentence.
There are a whole bunch of more similar rules, but they are all rather specific as to the syntactic situation that they are talking about. If the rules were simplified or made too general, then there would be too many so-called exceptions. Native English speakers know these rules implicitly (their ear does all the work), but they would usually be hard pressed to explain them--and if they attempted to explain them, their explanations would often be wrong and/or misleading.
Best Answer
I suppose it's fine if the narrator goes on to explain how the lying started, and uses the "when" answer to let the reader know the context. For example: