My Xbox One S is showing 365 GB of storage even with a 1TB SSD drive installed.

We cannot assist with self-repair.

Xbox Series X|S games can only be installed on the console's internal storage or the external Seagate SSD card. Link here. External drives are too slow for those games. The external hard drive can only be used to install backward compatible games (Xbox, Xbox 360, Xbox One).

Also be aware, that you don't get the full amount of storage on the internal drive, as the console's system and other system related storage needs are deducted from the drive space.

So even if the packaging advertises a 512GB or 1TB drive in the console, you don't get the full 512GB or 1TB of storage. So don't worry, if the console shows less space. So if your console shows you something like 800-something GB, then you have a 1TB console,
and if you have 400-something GB you have a 512GB console.

You can however simply add up to three additional external USB3.0 hard drives to get more storage space.

My guess would be a 1TB SSD minimum. Despite prices dropping on SSD's, they're still very expensive per GB compared to old traditional mechanical hard drives.

On my PC I recently upgraded to an internal 2.5" 2 TB SSD drive for additional game storage. It wasn't cheap, coming in at just under £200 ($258!). Now SSD prices will come down in the next 10 months or so, but by how much? I'm not sure. I think a 2 or
3 TB internal drive at launch is a lofty expectation.

I installed it on my Crucial MX500 1TB SSD. I wouldn't have even asked if I installed in on the Hard drive. The hard drive is just archival stuff for records and family pictures. I don't put any games on it.