Violetta,
It's so painful to read this thread because I was going through the exact same frustration. I was reading far to many posts and no one had the answer. My experienced electrician was baffled. He aggressively suggested I simply use the mechanical switches that worked. I was far too stubborn to let this one go.
The Lutron switch is different than a mechanical switch. The Lutron switch is digital. Many electrical engineers do not know this because digital switches are new. So you can have an electrical engineer with 40 years experience who is completely baffled.
I never heard if your situation was a single pole setup or a 3-way. Since you never mentioned 3-way or an additional switch location, I assume you have a single pole setup.
In a single pole scenario...(this one is not my experience)
Try switching the two wires.
In a mechanical switch, all you are doing is completing a circuit. Very much like allowing two wires to touch. In a digital switch, it's not mechanical. So how would you have the two wires connect without a switch OR WITHOUT POWER? I believe you have to have power to the black terminal and it may not be the wire you think it is. Once you have power to the digital switch, it will allow you turn on the light.
In a 3-way scenario...(my experience)
On a mechanical switch, it does not matter which traveler wire is connected to a traveler terminal. On the Lutron switch, it does!!!!! For example the white wire should go to the same terminal on BOTH switches, like the the blue screw terminal. The best way to test this is with a continuity test. A clamp meter can test this for you.
The problem for me was even super difficult. My nightmare with my 3 switches (which had 3 more corresponding switches on the other side of the room.) had one switch with black as the common, the next one had red as the common and the last had the 2 traveler wires crossed. The red wire at one switch was DIFFERENT than the red wire at the other location. At this other location, it was actually the white wire. NIGHTMARE!!! This is why I believe everyone should NEVER trust the color of the wires. Simply buy a clamp meter ($20 or $30 bucks).
This is the video that cleared it all up for me...
How to Fix a Digital 3-way Switch System (Multi-location Dimmer Switches)
https://www.youtube.com/watch?v=KV0UAa5m-wM
Thanks for listening,
-Clifton Grappler
Always remember; Nowadays, YouTube always has the answer! LOL