An appraisal contingency means that the buyer can back out of the transaction without losing their earnest money deposit if an appraiser determines that the home is worth less than the offer price. It is your decision whether to include an appraisal contingency. Therefore, if you waive the appraisal contingency with the original seller, you will also need to waive the inspection contingency when purchasing back from Better Real Estate.
Including the contingency will make your offer less competitive in some markets. If you do waive the appraisal contingency and the appraisal comes in below purchase price, you will be responsible for any changes in the loan terms, including the monthly payment, mortgage insurance, or down payment. If you cannot meet these, you will not be able to purchase the home, but will be refunded your earnest money deposit less the cash offer transaction fee and any transaction related costs.