I got it:
In the config for your stores, just enter the URLs as they are, not the install URL. Then, on the alternative domain (otherstore.com), use the default htaccess (when it is in root, otherwise add rewrite in htaccess)and put just this inside the index.php:
require __DIR__ . '/app/bootstrap.php';
$params = $_SERVER;
$params[\Magento\Store\Model\StoreManager::PARAM_RUN_CODE] = 'websitecode';
$params[\Magento\Store\Model\StoreManager::PARAM_RUN_TYPE] = 'website';
$bootstrap = \Magento\Framework\App\Bootstrap::create(BP, $params);
/** @var \Magento\Framework\App\Http $app */
$app = $bootstrap->createApplication('Magento\Framework\App\Http');
$bootstrap->run($app);
The following is key though, i did not get it to work otherwise, au contraire Magento 1 where we had lots of stores without this. Go to your other store root with ssh and enter this:
ln -s /path/to/your/install/public_html/m2/app/ app
ln -s /path/to/your/install/public_html/m2/lib/ lib
ln -s /path/to/your/install/public_html/m2/pub/ pub
ln -s /path/to/your/install/public_html/m2/var/ var
After we added the latter, everything seems to work fine, so far.
Good luck!