Save Photo's library on Icloud but accessible with Mac and Iphone

On iCloud servers is correct. So pictures can be downloaded or printed from any browser. The full sized images are also saved anywhere you have "Optimize" un-checked and "Download Originals" is checked, so you can keep the full library on an external hard drive. (If you have "Optimize" checked, then a "backup" of that library may only get low resolution images.)

If you use an "Optimized" Library, then when you look at the pictures on your computer or phone, you get screen-sized images. But when you go to Edit mode, or you choose to print, the full sized images are downloaded quite quickly from iCloud. You usually don't notice any delay at all. Actually, the process is dynamic, and full sized images are stored locally until the extra space is needed.

I have "Optimized" checked on my phone and iPad, but "Download Originals" checked on my Mac, since I have the room there. I use Time Machine to back it up, and I make copies on a separate drive monthly, more or less. Remember, iCloud is not a backup. If you accidentally delete pictures on your phone (and delete from "Recently Deleted") they are gone from iCloud, too.

I actually keep two Libraries (well, really more)-- one with just favorites and recents that I sync with iCloud, and one much larger one that has all (kinda) of the pictures, and which is not connected to iCloud.

If I understand you, what you're asking is exactly what iCloud Photos does. iCloud Photos is not a backup service, but rather it is a synchronization service that keeps Photos on all connected devices in sync. If you take a picture with your iPhone, that picture is uploaded to iCloud and downloaded to your macs and your iPad and your iWhatever. If you delete a picture on your Mac, it is then deleted from iCloud and your other devices. And you can access Photos from any computer by going to iCloud.com. Here is some information about setting this up:

Set up and use iCloud Photos - Apple Support

In Photos on you Mac, you can choose "Optimize Mac Storage" to keep only screen sized images on you computer, saving as much as 85% of the space, depending on use. Or you can keep your Library on an external hard drive where you might not want to use the Optimize option. An external hard drive, though, needs to stay connected, because Photos does lots of processing in the background.

If Optimize isn't used, then the Library can be backed up by just copying it to another location from your hard drive. You can also use Time Machine or Carbon Copy Cloner, for example, to do incremental backups.

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