For the Contact Form 7 you show [anr_nocaptcha g-recaptcha-response] but for a custom form you suggest [anr-captcha]. I think it would be helpful to describe both shortcodes (and why they are different) and to talk about the g-recaptcha-response parameter.
[anr_nocaptcha g-recaptcha-response] is not a shortcode. It is just a CF7 tag which will work only in CF7 form. g-recaptcha-response is the name of the field. To know more about it please see https://contactform7.com/tag-syntax/
[anr-captcha] is WP shortcode which will output the captcha html.
Many thanks for the clarification!
