Does not the 'me' in Nimeweka make it 'I have put' ?
Yeah. It's not an exact exact exact equivalence, but for the purpose of this course, they should be consistent with it. Report it if it comes up again.
Agree with Jimmie
Why not ‘I have put the cup on the table’? I have reported it.
Why is "I put a cup on the table" rejected? Does something in the sentence suggest any specific cup?
No, but I think your problem may be in saying "put" instead of "have put". "-me-" is present perfect, not normal past tense.