לועזי would be on the "foreign" calendar, i.e. the Gregorian calendar.
For example, if your child was born today, their Hebrew birthday would be the 15th of Iyar, and their "foreign" birthday would be the 27th of April. Next year, their Hebrew birthday will still be on the 15th of Iyar, which will fall on the 16th of May, and their "foreign" birthday would still be on the 27th of April, which will fall on the Hebrew date of the 26th of Nisan.
יום הולדת לועזי is a shorter form of יום הולדת על פי התאריך הלועזי.
תאריך עברי / תאריך לועזי mean date according to the Jewish / gentile (Gregorian) calendar. לעז means non Jewish/Hebrew.
See for example בְּצֵאת יִשְׂרָאֵל מִמִּצְרָיִם בֵּית יַעֲקֹב מֵעַם לֹעֵז (Psalms 114:1).
Both dates intersect every 19 years I believe.
If one has a Muslim father and a Jewish mother, and dwells in the West, ze can celebrate birthday about thrice a year.