Under international law white phosphorous is considered an incendiary weapon, not a chemical weapon. There is a list of chemical weapons and white phosphorous is not on that list, although it is often incorrectly claimed to be so.
The Chemical Weapons Convention, sometimes invoked in discussions of WP usage, is meant to prohibit weapons that are "dependent on the use of the toxic properties of chemicals as a method of warfare" (Article II, Definitions, 9, "Purposes not Prohibited" c.). The convention defines a "toxic chemical" as a substance "which through its chemical action on life processes can cause death, temporary incapacitation or permanent harm to humans or animals" (CWC, II). An annex lists chemicals that are restricted under the convention, and WP is not listed in the Schedules of chemical weapons or precursors.
The fact that an American intelligence analyst once during the Gulf War miscategorized WP as a "chemical weapon" when Saddam Hussein used it doesn't change its listing under international law.
Your 'key quote' misses the next section, which makes my case.
> No it's not forbidden by the CWC if it is used within the context of a military application that does not require or does not intend to use the toxic properties of white phosphorus. White phosphorus is normally used to produce smoke, to camouflage movement.
> If that is the purpose for which the white phosphorus is used, then that is considered under the convention legitimate use.
> If on the other hand the toxic properties of white phosphorus are specifically intended to be used as a weapon, that, of course, is prohibited, because the way the convention is structured or applied, any chemicals used against humans or animals that cause harm or death through the toxic properties of the chemical are considered chemical weapons.
~ Peter Kaiser, spokesman for the Organisation for the Prohibition of Chemical Weapons
Dropping white phosphorus during the daytime (when the illumination features of WP aren't needed) on a school in a dense urban environment is pretty clearly the case where it's considered to be a chemical weapon.
The annex lists chemicals understood to be chemical weapons under the convention, but it doesn't purport to be a complete list, only a set of some examples.
Er, but unless I missed something, Amnesty International didn't claim Israel used it for its toxic properties against humans or animals? They were against it being used for any purpose in Gaza, and chiefly cite it causing property damage. While that's still bad, it's not true that in that case WP counts as a banned chemical weapon. Ergo, the claim is false. White phosphorous is not a banned chemical weapon, and was not used for its chemical properties against humans or animals.
Also, even Amnesty International mentions that the purpose wasn't "illumination" — which would be ridiculous — the purpose was camouflage (which is apparently a typical use case).
The Amnesty International article is against any use in Gaza _because_ any use in a dense urban env is by definition an illegal use of chemical weapons.
Camouflage isn't a valid use of chemical weapons on a dense population when the arguments are taken as a whole. Even Mustard Gas works as a camouflage, it'd mean the whole convention is pretty much unenforceable.
Even the Amnesty International article doesn't call Israel's usage of WP as using a chemical weapon: they call it an incendiary weapon in the very first sentence (which goes along with the Wikipedia article I linked). The Mustard Gas comparison isn't valid; the whole point of mustard gas is to target humans. WP was being used as camouflage, and caused property damage. That can be bad! That can even be a war crime. But being bad or a war crime doesn't mean Israel used chemical weapons against people in Gaza. They used an incendiary weapon that caused a lot of property damage (which is possibly a war crime in its own right).
Edit: the entire Gaza War started with Hamas firing rockets packed with incendiary chemicals into dense Israeli urban environments. I would say many things about Hamas, but I would not say that Hamas has used banned chemical weapons against Israel. I suppose if you believe Hamas has used banned chemical weapons against Israelis, you may certainly believe Israel has done the same. But I think your definition of chemical weapons is significantly broader than international law. AFAIK neither side has done this.
The event they're talking about killed 2 and critically injured 14 more when WP was used over an in use UN run school. I'm not sure how you attached to the property damage component, but it's not a core piece of the argument being made by me or Amnesty International.
Edit in response to your edit: It's not the incindiary part of WP that's an issue, it's the chemical toxicity to humans and animals. Hamas has not used anything approaching chemical weapons by any definition against Israel, despite your attempt to deflect.
https://en.m.wikipedia.org/wiki/White_phosphorus_munitions
Key quote:
The Chemical Weapons Convention, sometimes invoked in discussions of WP usage, is meant to prohibit weapons that are "dependent on the use of the toxic properties of chemicals as a method of warfare" (Article II, Definitions, 9, "Purposes not Prohibited" c.). The convention defines a "toxic chemical" as a substance "which through its chemical action on life processes can cause death, temporary incapacitation or permanent harm to humans or animals" (CWC, II). An annex lists chemicals that are restricted under the convention, and WP is not listed in the Schedules of chemical weapons or precursors.
The fact that an American intelligence analyst once during the Gulf War miscategorized WP as a "chemical weapon" when Saddam Hussein used it doesn't change its listing under international law.
So, yes, the claim is false.