Better words for "open-ended" and "multiple choice"?

For the "multiple-choice", you could use "enumerated"; however, it probably makes more sense to just list the options, using a full phrase: "either true or false", "one of: json, xml, text", etc.

For the "open-ended", you could use "free-form" or "arbitrary text"; but it's likely that there are some restrictions, in which case you can probably just describe those (e.g., "up to 128 characters", "cannot include whitespace") and leave it implied that there's not a fixed list of options.

You could draw on the relatively familiar idioms of Web forms and describe the multiple-choice parameters as selections and the open-ended parameters as text inputs.