It's all marketing. They are probably all right, it's just the way to interpret them.
Airbus have sold fewer A320 family planes then Boeing has. But the 737 has been around a longer time, so it's safe to assume that more 737s have been retired (notably 100, 200 and 300 series). Therefor right now there might be more A320s flying around then there are 737s.
Also, note they say "most popular"! Not most numerous. So above argument could be moot. They might have send a questionary to 100 people asking which plane they liked best. The Airbus might have come on top of the questionary, but that does not mean anything. After all, who books their flight ticket based on what plane they'll get? Except for plane geeks noone does that anymore. back in the day of Concorde perhaps, but not today.
Boeing's claim to have delivered 7100 planes I won´t comment on. Such a statement is easily verified, and therefor probably true. More interesting is their claim at having sold 10800 respectively 9700 planes. Here the 9700
planes might refer to actual confirmed and signed orders, whereas the 10800 figure (which I can't find on the Boeing site) might refer to actual orders and
letter of intents and options which are not confirmed but for which the negotiations are largely complete.