@Russell_J
The search widget has been modified to use full text indexes in the November update of ArcGIS Online and since then, doesn't work as it should in existing maps. Searches with 'contains' work as 'starts with' or just don't return anything.
This is supposed to be solved by saving the map so it's upgraded and takes advantage of the new search function. I could successfully do this with a few maps, but not all.
Another way is to use the addToDefinition method on the REST point of the hosted feature layer. I've encountered 2 issues with this option:
1. this method doesn't work if the layer has related views (see map 'Erfgoed' and layer 'Domein', search is set on field 'Domein' with keyword 'Contains').
2 even if the full text index is added, the search still doesn't work properly (see map 'PFAS' and layer 'domein_PFAS', search is set on field 'Domein' with keyword 'Contains').