Turgon's people consisted of both Sindar and Noldor. I'm assuming they spoke Sindarin as a matter of course, and probably Quenya for purposes that concerned strictly the Noldor. So yes, Gondolin would have been the commonly used name and Ondolinde the one he heard less often. I think Ondolinde is easier to distance himself from - especially because no one refers to him in lays and tales as "Laurefindil of Ondolinde". He'd much rather be Glorfindel of Imladris now, I think. Dying was hard enough - learning to live with it is harder still.