I created a sign off page and generated a dummy signature for it. For years, our process has been to click on sign, draw a rectangle to encapsulate the signature, then enter the passcode. The document, then gets stored on a secured server, picked up by the next person and so on. Some documents come back from, scanned, from the field and are reviewed. Each section requires a signature. Currently, this has become a three pass procedure where we use the caligraphic "signature", on each section, go back and make the document editable, then add the date to each "signature", then sign.
At no time have we constructed any kind of field. I could find nothing about certifying a document, other than the verbage inside of the newer signature dialog box. Not sure about different types of signatures.
The signature in your SignedSignature.pdf has both a DocMDP and a FieldMDP transform; in other words, it's a certification signature with even more restrictions. So in particular after that signature others are not allowed to add new signature fields, you have to prepare empty signature fields before certifying to allow others to countersign.
So apparently where you "click on sign" you have a functionality that creates a certification signature. And indeed, the bottom-most icon in the toolbar screenshot you show is the icon for "Certify (visible signatures)". Instead of that you should use the "Digitally sign" functionality of the "Use a certificate" tool:

(You can change the customization of your toolbar to contain that functionality in addition to or instead of the certify functionality.)
I just signed your Signature.pdf with "Digitally sign", see the attached Signature-Regular.pdf, and this file can be signed arbitrarily by others.