If you're asking whether you need a lawyer to draft a contract, legally, the answer is no. Anyone can draft a contract on their own and as long as the elements above are included and both parties are legally competent and consent to the agreement, it is generally lawful.
To write a simple contract, title it clearly, identify all parties and specify terms (services or payments). Include an offer, acceptance, consideration, and intent. Add a signature and date for enforceability. Written contracts reduce disputes and offer better legal security than verbal ones.