I want to make the tool-tip reads its content from a specific field of the item template, as I have a rendering variant that contains many fields of different templates.
Search Results rendering is using Rendering Variants to display results. If you want to modify how HTML of particular search results item is generated then you would need to override Rendering Variants logic (that might be pretty hard). We are using Field Renderers (Sitecore is generating HTML) there like that:
FieldRenderer fieldRender = new FieldRenderer
Item = item,
FieldName = variantField.FieldName,
This HTML is later on put on the fronted into
<ul><li> list. And as far as I can see in the code we are not adding any tooltips there. So the tooltip comes from Field Renderer probably.
Beside all above you can try use Rendering Variant tokens: