OK, so you do have iCloud Photo Library turned on. That means that you have uploaded all the photos in your Camera Roll up to iCloud, and all new photos go there as well.
However, with iCloud Photo Storage, the entire library is housed on both iCloud and your device. If you select "Download Originals" then all of the photos that can fit on your device will be on the device locally. If you select "Optimize Photos", then as many originals that can fit within the phone's storage limit will be on the phone, and any other photos will be put on the phone as thumbnails only, which cannot be viewed when not connected to Wifi.
So, depending on the size of your library, if everything can fit on your device as originals, then they all will be available offline.
One thing to remember is that iCloud Photo Library is a sharing service, not an archival one. What that means is that all of the photos go to the library in iCloud, and then all of them are pushed back down and shared with all devices that are signed into that same iCloud Photo Library. Anything you do to those photos will be reflected everywhere. So, if you delete a photo on your phone, it will be deleted from your Library completely.
iCloud Photo Library FAQ - Apple Support
iCloud Photo Library Help - Apple Support
If you want your photos stored in a safe place, you should either import them to the Photos app on your computer, or put them in an archival cloud service like Dropbox or Flikr:
PHOTO IMPORT IOS TO MAC/PC
If you only want selected photos from your Library on the phone, you have two options, but both require that you turn off iCloud Photo Library on the phone:
1. Sync photos to the phone via iTunes. If you have your photos imported to the photos app on your computer, you can use iTunes to sync selected photos back to your phone:
iTunes 12 for Mac: Sync iPod, iPhone, or iPad
iTunes 12 for Windows: Sync iPod, iPhone, or iPad
2. You can also put selected photos on your phone by using iCloud Photo Sharing:
iCloud Photo Sharing - Apple Support
Get help using iCloud Photo Sharing and shared albums - Apple Support
But remember, if you decide to go either of these routes, you would need to turn off iCloud Photo Sharing on the phone.
Post back if you have any questions!
Cheers,
GB