The home page search bar comes from the Visual Composer element, so terms like “All Activities”, “All Range” etc can be changed in the VC itself as shown in the “Example” underneath the search box. For example, you have
Prezzo::price_range
and you can change this to
Prezzo::price_range::My Label Here
And where it says My Label Here change that to your custom Label to show in frontend instead of the default “All Range”
Please see the attached screenshot for where to edit it.