There are two sides to the £2 fare. The official Government line is that it is to help with the cost of living, under the
Help for Households scheme. All of the official promotional material has that logo and is in the same green colour and style that the other schemes use. They are aiming it directly at cutting the cost of travel.
The big bus operators (First, Stagecoach and GoAhead at least, we don't have any Arriva locally so I'm not seeing their communications) on the other hand see this is an opportunity to market bus travel with the "give it a go, its just £2" type of message. They are looking to grow the market.
The Communications team at the local authority I work for takes the line that we should follow Government policy, so all of our publicity has followed the Government "Help for Households" line, rather than the "try the bus" angle. As such, it has been covered in our weekly residents' email newsletter, is on both the main council and travel planning websites, has been regularly Tweeted / Face booked etc from both brands, appears on the Cost of Living Help section of the council's website, and on the front page of the travel planning site. The local press has also picked up on it from our council news release. The Communications team view of posters (we don't have much Real Time coverage) at bus stops was that existing passengers will be pleasantly surprised when boarding, but potential new passengers won't, generally, be at a bus stop. So digital has been our way, using existing council channels.
I've yet to see any patronage figures, but haven't heard of any panic among the team trying to deal with overloading.