A field of type "Project Specific Select Field (simple)" is technically a text field, where you can set the value as described above.
"customfield_80000": {
"value": "red"
}
But fields of type "Project Specific Select Field" are technically select fields like fixVersion and component field. These are set via ID and not via value.
"versions": [
{
"id": "10000"
}
]
Please have a look at the REST API.