You aint sure what the lie was? The lie was that it wasn't his fault. He created the situation with his policy change which Sessions sent to the DHS. These are facts that will not be explained away by some kind of whataboutism as it pertains to the laws on teh books and who is responsible for those laws.
Did some kids get separated in teh past from their parents? YES, if their parents were determined to be MS13 or terrorists or whatever, but a complete and no tolerance policy, enacted by the POTUS, lead to the prosecution of ALL the of age parents, and thus their separation from their children.
When he told the American people that he couldn't do anything about it, he lied. When he claimed it wasn't his fault, he lied. Theres more lies, but those are the important ones.
The situation sucks, people are trying to come into the USA illegally. Locking them up and separating them from their children is a tough guy act that Trump wanted to do as a deterrant. He actually is upset that he had to back down from it. They were opening up more child tent camps in anticipation of more. Since May 2k kids have been separated, this whole situation was created to make a political crisis. It was planned, it was deliberate.
This "policy" you refer to, is nothing more than Trump telling Sessions to uphold the existing law. Trump has repeatedly said that congress needs to change the law. Changing the law is a non starter for dems and some repubs in the senate; by their own statements. So who is ultimately responsible; Logically? The dems and some repubs, true colors will be revealed now that Trump has signed the EO. Keeping the children with their parents will not be enough for them, because their true desire is simply let these people walk into country unimpeded. Just watch and see.