Is it ever imperative or more correct to say, e.g.,
"I heard that he won the lottery" vs. "I heard he won the lottery" or
"I have trained in surgery" vs "I trained in surgery"
"I have had a positive impression of him" vs "I had a positive impression"?
Is it ever imperative or more correct to say:
"I heard that he won the lottery" vs. "I heard he won the lottery"
Only in formal writing would I see it as better to include the 'that'. In thinking about my conversational speech, I'm as just as likely to include the 'that' as not.
With:
"I have trained in surgery" vs "I trained in surgery"
"I have had a positive impression of him" vs "I had a positive impression"
You're actually asking, is it OK to use the Past Tense (no 'have'") INSTEAD OF the Present Perfect Tense!
Which tense is used depends on the speaker's view of a situation.