While it is "allowed" - the CDC just wants you to get whatever shot is in front of you, period - there is conflicting information about what is "best", because we quite simply don't know with any certainty.
That's why many places are recommending that you stick with the same one, even if only because if you did not have a reaction to it the first two times, chances are on shot three you won't have any adverse reactions, either. Major vaccination providers like Walgreens, for example, will only let you make an appointment at a site that has the same type you already got (which is why it wanted to send me like 45 minutes away). I do know that when I finally was able to get one at a mass vaccination site (I gave up on Walgreens), there were multiple lines - they took a look at your card, and sent you to the line that corresponded with what you previously got.
So I think it's a little harsh to declare the pharmacist was "spreading false information".