Yes sorry William, I just looked up Pusley again , but it seems there are several different types. Pusley seems to be an alternative name for Purslane. Florida Pusley seems to be an annual, native just to Florida (there's a surprise!), whereas Purslane is a perennial. But they are sometimes found growing together! Checking the scientific names they belong to different families as you have rightly pointed out.
I imagine the only way you will get a definitive answer on whether it is rabbit safe, is to find information from a local source.
Reminds me once again that it's always safer to use the scientific name.