Yes, but Alexandra was born in Germany. Not in Russia.
That's the complicating issue. If Alexandra had remained in the west instead of moving to Russia, she would have continued to celebrate her birthday on 6 June, and the enitre issue of old style equivalency would be irrelevent. So in that sense, it's hard to say her insistance on observing the 6th as her "real" birthdate is mistaken.
The whole thing is confusing beyond all reason, isn't it?
Basically, if you want to observe the technically correct date according to the calendar, use the 7th. If you want to observe the same day Alexandra herself preferred, choose the 6th instead.