Unfortunately, I don't have time right now to develop this in too much depth or nuance, but it's a very interesting question and I want to give you a bit of an answer!

The very brief answer is that the Germans used the term rex (Latin for King) to refer to rulers of various peoples to their east only if the German King/Emperor had made some individual from that ruling family a king. One of the conditions that would lead to this was definitely that the original person who was crowned king was a Christian. Another was that the individual being crowned delivered some benefit to the German monarch. So, the Piasts (ruling house of Poland) Christianized much earlier than the Lithuanians, and the first person who was crowned king/rex, Boleslaw I, basically forged an alliance with Otto I of Germany and Otto sealed it by crowning Boleslaw. This is what happened with the Premislyds (the Czech ruling family) and the Arpads (the Hungarian ruling family).

Some points that this raises:

-- The reason this was an option in the first place is at least partially linguistic. You're right that, for example, the Dukes of Lithuania, or other rulers such as the Princes of Hungary, etc. are basically monarchs. De facto monarchs that weren't called kings aren't exclusive to Central and Eastern Europe. But in these cases which you seem to be thinking of, what is going on is that you're seeing a mismatch of what is essentially technical vocabulary. This is still an issue, for example trying to translate Graf v. Landgraf v. Freiherr, etc. into English from German can be hard, because there isn't a one-to-one correlation between noble titles. The same with German and the Slavic languages. The Germans basically took rulers that basically were kings, and just said "no, no, no, you aren't a rex until we say you are."

-- Why did the Arpads, Piasts, etc. care? Because the kings of Germany, not to mention other Christian rulers had a cache, and because trading with those areas, having military alliances with them, etc. was something they wanted. So Christianizing and then getting the German ruler to make you a rex was a way to start an alliance with the Germans that would bring benefits. Technically it meant that the new king was subordinate to the German one (who was obviously more powerful since he was a kingmaker). However, there was little centralized control at this time, so it wasn't that extreme of a commitment. Conversely, the German kings were willing to make other kings because a) it proved they were boss kings b) it was a way to get influence over those other kingdoms.

-- Christianity seems to be a pretty important component of this? Yes! Partially because Christians weren't supposed to play nice with others until those others became Christians, too. But there was a bigger force working, that of sacral kingship. That's a huge topic, but basically there are a lot of religious issues tied into being the king (this is pretty universal, actually). So once you become a Christian, you need to realign to the Christian version of sacral kingship, which was culturally bound up with western European ideas about what kings were. One way to do that it so get yourself declared king by the Holy Roman Emperor.

I would point you toward Gabor Klaniczay's Holy Rulers and Blessed Princesses for more on kingship in that part of Europe, and how they established themselves as Christian kings. I'd also point you toward Francis Oakley's Empty Bottles of Gentilism for a look at sacral kingship.