Creating a Signature Field on a form?

They're probably referring to the signature appearance by which, if they do have Adobe Reader DC or Adobe Acrobat Pro DC,  they can customize their own signature appearance via the GUI provided by the signing action when they click on the signing field.

I assue that by cursive they mean an imported graphic of their wet signature., and like I mentioned, this is something they can do at signing time when they create their own certificate-based signature; you need to include that signature field for them in your PDF.

The other option is to use JavaScript scripting on that signature field  but this is very complicated, and I don't think that in this case this is the way to go.

It may be that they could also be referring to when they open the PDF they are doing it from a web browser which doesn't necessarily means that it will render properly; thus they're not able to click on the signing field and the popup dialogue window doesn't show with web browsers.

So, to recap, the text box that they're referring to may be the signing dialogue window which only appears if you've provided a digital signature field. This is done with the "Certificate" tool in Adobe Acrobat, not the "Fill & Sign" tool.

If you already provided a digital signature field, confirm with your Client  in which type of program are they viewing this PDF. If they're using a web browser encourage them to use Adobe Reader DC which is free to download, install and use.

If they refuse to use Adobe Reader DC and prefer to use a web browser then send the PDF via the Adobe Sign tool which is featured in Adobe Acrobat Pro DC. Using this alternate workflow will allow them to sign from a web browser or a mobile device without the need to install Adobe software in their computers nor subscribe to a paid plan of Acrobat Pro DC.