Help - I have two different E-signature fields

When creating forms you have the choice of creating them as normal pdf forms for use by arbitrary pdf processors or as forms for use with Adobe Acrobat Sign.

It looks like the first form was created using the former choice while the second one was created using the latter choice. 

In detail:

When starting to create a form, you are at a screen like this:

If you want to create a normal PDF form, don't check the "This document requires signatures" checkbox! The checkbox title is misleading, it should have been something like "This document is for use with Acrobat Sign".

If you have created a form with that checkbox checked, you can still revert it to a normal form, click on "More" in the panel on the right:

Now select "Revert to Acrobat Form". Be sure to test your form fields thereafter, AcroForm forms and Acrobat Sign forms are different, so changing the type might damage some functionality.