The correct information *is* reflected.

Again , I think there's just a fundamental misunderstanding of what you're looking at on this website. The "Scw’exmx Tribal Council" label that you see is just arbitrarily rendered near this spot when you look at the website, because geographically it's happens to be near the centrepoint of all of the other reserve lands that belong to the other member bands of the STC. There is no label deliberately placed here. 

When you look at https://www.openstreetmap.org/, move your mouse cursor over to the right side of your screen to the fourth icon from the top, labelled 'layers'. This will open a pane on the right side that allows you to pick another 'style' of map to see. Pick "Cycle Map", "Transport Map" or "Humanitarian" and *POOF!* the STC label is gone. That's because those other 'layers' are programmed such that they don't even bother rendering the boundaries of native reserves at all.

The "Standard" layer is the aforementioned "Carto" (https://wiki.openstreetmap.org/wiki/OpenStreetMap_Carto); it's developed by a third-party. It's just a 'style' that the underlying map data is rendered in. Having the "Scw’exmx Tribal Council" label here means NOTHING: it's just how Carto happens to render it. It's happenstance. If you don't want to see "Scw’exmx Tribal Council" on the map just pick a style that doesn't display it.

Otherwise the only way to get Carto to not display the STC label is to start playing around with the boundaries of this and other reserve lands in BC to 'spoof' the Carto renderer into thinking the label ought to go somewhere else, or simply delete the UNB from the STC, neither of which anyone here is going to do for you. That would violate a mapping rule we follow (https://wiki.openstreetmap.org/wiki/Tagging_for_the_renderer), and it would be factually incorrect.