The best English translations say “regret” and the others use the synonyms you mentioned so I’d accept what the Rabbi said about the translation (I personally don’t know Hebrew btw).
Remember God has emotions & He’s Holy, unlike us fallen creatures, so what it means for God to regret is not the same as human regret.
It shows God truly lamented like Jesus when he heard John the Baptist was killed, of course God knew John was going to die – he ordained it himself in the past – but when it actually occurs he still has an emotional response. It only means that God also foreknown emotional suffering & that he truly cares about us. He knew when he ordained it in eternity past that he would experience the feeling of regret or sorrow, and he was willing to suffer for the greatest good.
Remember Romans 9:22 – 23 What if God, desiring to show his wrath and to make known his power, has endured with much patience vessels of wrath prepared for destruction, in order to make known the riches of his glory for vessels of mercy, which he has prepared beforehand for glory—
John Piper had a neat Q&A on this I will share it on Youtube.