Bert:
First, you are not creating partitions by creating a hierarchy in the key name. This is using the 'default' cache provider (or partition) to store these cache entries. Each partition is a separate derivative instance of the xPDOCache class, e.g. xPDOFileCache, xPDOMemCache, xPDOAPCCache, etc. that has a key and a configuration.
That said, I recommend getting a cache partition first, and calling delete() directly on it. In your attempt, you are not specifying the partition to remove that cache entry from, so it is attempting to delete it from the 'default' cache partition. You can provide the partition details in the modCacheManager->delete() call, or, as I said, get the specific provider first and work directly with it:
Here is an example of creating a custom partition and deleting a key within it:
/** @var modX $modx */
$cacheKey = $modx->getOption('cache_customcmp_key', $scriptProperties, 'customcmp');
$cacheOptions = array(
xPDO::OPT_CACHE_HANDLER => $modx->getOption("cache_{$cacheKey}_handler", $scriptProperties, $modx->getOption(xPDO::OPT_CACHE_HANDLER)),
xPDO::OPT_CACHE_EXPIRES => $modx->getOption("cache_{$cacheKey}_expires", $scriptProperties, $modx->getOption(xPDO::OPT_CACHE_EXPIRES)),
);
/** @var xPDOCache $customCache */
$customCache = $modx->cacheManager->getCacheProvider($cacheKey, $cacheOptions);
if ($customCache) {
$contextCache->delete('noticeboard/list/' . $item->get('id') . '.detail');
} else {
$modx->log(modX::LOG_LEVEL_ERROR, "Could not get custom cache partition with key {$cacheKey}");
}