The earliest biography of Muhammad was written about 100 years after his death, so any historicism is on slightly shaky grounds. None of these stories would be accepted by all Muslims -- there is a lot of room for plausible deniability. I don't think this question is answerable in the form "did Muhammad really do this" or "would all Muslims agree that Muhammad did this?"
But the real question you're wondering about is probably more like the following: how many of these deaths were attributed to Muhammad by Muslims soon after his death, for the reasons given in the article?
The answer: all of them.
Where you see the words "Sirat Rasul Allah" on the list of primary sources in this table, this refers to Ibn Ishaq's early biography of Muhammad, which is the source for probably 90% of any biographical information anyone has ever found about Muhammad. Muslims often doubt the historical veracity of the stories in this book: for example, it contains poetry ascribed to people who were not poets, and the Satanic Verses story which many Muslims discount. But non-Muslims should be in a state of doubt about everything said about and attributed to Muhammad, and this "Sirat Rasul Allah" was indeed written by a Muslim, and much of the book has been reused in other biographies.
A typical example of the story behind the WikiIslam article can be found in the Wikipedia article on one of the first deaths named, Abu 'Afak. Ibn Ishaq's "Sirat Rasul Allah" says that Muhammad ordered this killing himself. Another account, by an historian named Al-Waqidi, says that one of Muhammad's followers killed Abu 'Afak on his own initiative.
One can debate endlessly over which of these stories is more plausible. There can be no resolution to that. But did an early Muslim, and an intelligent historical writer, state that Muhammad directly ordered the killing? Yes. So it was within the range of plausibility in the early Muslim community.
Either way, no one claims that anyone was admonished for this murder of a poet.