As you say, the first time that they meet each other after their initial meeting (where Claire steals form Ray) is in Rome. They have a genuine conversation, where Ray accuses Claire of the theft and she vehemently denies it.
Eventually they get together and fall in love, and hatch a plan to become rich by stealing corporate secrets from either a pharmaceutical company or a frozen pizza company. They eventually decide to go with the pharmaceutical company, as they think that their hair growth product will ultimately provide them with a larger payoff.
Once they decide that the company that Claire works for (the pharmaceutical company) will offer the highest value secret, Ray then gets a job in their largest rival pharmaceutical company. Claire then acts as a turncoat for her company as if she is to turn over the secret to Ray's company, when really they are colluding to steal the secret together using the resources from Ray's company, and they will then sell the hair growth formula to a third company for personal profit, as opposed to helping the rival company.
In order for this plan to succeed, no one can know that they know each other personally. They must act as if she is a true turncoat, but he needs to be there in the rival company to be the primary contact, so that they are the only people who have access to the secret, meaning that they are the only people who profit from it.
Considering that their first meeting happened when they were both working for government institutions, others may have had knowledge of the fact that they had met before, particularly because everyone else in Ray's corporate counter-intelligence department was probably an ex-spy. Therefore pretending that they had never met before wasn't an option, as that cover could have easily been blown.
As they needed to pretend not to know each other, this means that they would not have met in Rome and fallen in love. What better way to feign their first meeting (as they were being monitored at the time) by recreating their real first meeting? This would mean using the exact same topic of conversation, and almost the exact same words. It makes the entire encounter more believable, and would remove any scrutiny that may have followed them providing they had previous experience with each other.