Plugin settings multiline input field
You are close, the template loaded by the macro can be found in craft/app/templates/_includes/forms/textarea.html. It does have a value attribute, and a few others, for example:

{{ forms.textareaField({
    label:         "Submitted Subject"|t,
    id:            'submittedSubject',
    name:          'submittedSubject',
    instructions:  "Input your setting"|t,
    value:         settings.submittedSubject,
    rows:          4,
    cols:          50,
    errors:        settings.getErrors('submittedSubject'),
    maxlength:     250,
    showCharsLeft: true,
    autofocus:     settings.hasErrors('submittedSubject'),
}) }}


All the form macros are defined in craft/app/templates/_includes/forms.html.