"And finally, the court's conservatives used their 5-4 advantage to rule that suspects must break their silence and tell police they are going to remain quiet if they want to invoke their "right to remain silent" and stop an interrogation, just as they must tell police that they want a lawyer."
That was a sensible case, if I recall correctly. The "right to remain silent" means exactly what it says. It has further been extended to mean that if you tell the police you're exercising your right to remain silent then they have to call off the interview.
What this guy did was just to sit almost-silent in an interview room for a couple of hours until eventually the cop managed to prise a confession out of him (for what it's worth, the charge was murder and there was a lot of other evidence in addition to the confession, you can look it up). What the murderer's lawyer wanted was to have the confession deemed inadmissible (and hence a new trial, which was unlikely to result in a different verdict due to the overwhelming other evidence but would at least line the lawyer's pockets a bit more) on the grounds that sitting there and not answering questions for a couple of hours should oblige the police to stop asking questions.