Question about Tsar Nicholas the second's son?

Both are correct. Alexei is the Russian form of the name Alexis (which is of Greek origin). It used to be standard practice for writers in English to use the English forms of Russian names, so that, for example, Alexei's father was routinely referred to as 'Tsar Nicholas' rather than the Russian form, Nikolai. And Russian aristocrats very often spoke English among themselves and when they did they often used English name forms too.

The boy was christened Alexei Romanov

but sometimes referred to as Alexis in

written form.

Alexis and Alexei are both forms of masculine names

originating from the ancient Greek.