Dharma name or… simply ‘Buddhist’?

Sure, there is a traditional basis for “Dharma names” in Buddhism, in the Pahārāda sutta (AN 8.19)

Pahārāda, just as all the great rivers, that is to say, the Ganga, the Yamuna, the Aciravati, the Sarabhu and the Mahi, upon reaching the great ocean, losing their former names and origins, come to be known simply as the great ocean…

so, too, Pahārāda, when these 4 castes —the kshatriyas, the brahmins, the vaishyas, the shudras— having gone forth from the house for the homeless life in the Dharma-Vinaya proclaimed by the Tathāgata, they abandon their former names and lineages, and are simply known as ‘Sakya-putta recluses’ [‘recluses who are sons of the Śakya’].

But looking at this basis, you can see for yourself that, unless you live in a society where your name de facto informs others of your class / caste, you don't need a Dharma name!

Taking one would just transpose practices appropriate to the old caste system in India into another context, without regards for the appropriateness or priorities in the situation at hand (and “fixing imaginary flaws” is likely to distract you from, and keep you blind to, the present flaws —no society is perfect!).

Moreover, “sons of the Śakya” had no ‘exotic’ ring to it, in the regional context the Buddha proposed it! In Anglo-saxon countries, today, that'd be equivalent to changing your name to ‘Smith’: it lets your origins go, but it doesn't make you ‘special’, with some weird foreign name…

It might be noted that the Pāḷi term in “sons of the Śakya… … … …