I'm pretty sure this is actually wrong. Cullen saying "It seems you've become less of a friend to the templars in the years since your ascent" is based only on whether you spoke up in favor of Meredith at the start of Act 3 or not. If you do he says "I'm sure you know how much it meant to the Knight-Commander for you to side with her against Orsino." In both cases he refers more to the Champion's relationship with the Knight-Commander" and his own personal feelings are unaffected. You can still talk to him like he's an old friend and he's still super nice to you in his ambient and conversation dialogue. It would only be the case if he was much less friendly when you ask him how he is or click on him that the game would keep track of not being his friend.
So far I've never heard of anyone not being able to say "I'm wanted to see how an old friend is doing in this difficult time" or have Cullen not say "I hope my letter found you well" after he sends you notice of the rumors about Aveline.
To be clear, the "old friend" line I'm referencing is not Hawke calling him an old friend in that line everyone gets (shown in the video above), it's Cullen calling Hawke an old friend the first time you greet him in Act 3. And yes, he is polite to you regardless of considering you a friend or not, but then again so is Meredith. They maintain a certain profesionalism with Hawke, even if you go out of your way to antagonize them. That's why it's such an arcane trivia that no one remembers.
I can't tell you for sure what is the trigger for his different greetings, but I have got the "less of a friend" greeting even when siding with Meredith at the beginning of Act 3, and the "old friend" one even when siding with Orsino, so it can't be exclusively tied to that either. The most consistent results I've got have been with his personal conversations with Hawke, independently of Hawke's decision at the beginning of the final act.
In any case, as I said, that tile is more a blatant callout to Inquisition than to DA2. It's not really possible to establish whether your Hawke and Cullen were friends or not beyond your own headcanon, since they interact so sparsely during the game itself, so if anything, this tile is more asking how you want Cullen to refer/react to your Hawke showing up again, if he reacts at all.