The context is broader than that, but even if it hadn't been, it still doesn't change the meaning and intent of Yeltsin's demand. He wants Europe, and his argument is that Russia should be allowed to have Europe, because she is (a) European, and (b) powerful enough to protect the continent from future threats.
Yeltsin is so convinced this is a reasonable position that he tells Clinton the US can have all the other states, and protect them. He considers this a legitimate and entirely reasonable proposal.
Yeltsin also admits that Russia wants Asia, on the grounds that 'Russia is half European and half Asian.' He sees Russia's acquisition of Europe and Asia as inevitable, telling Clinton 'Eventually we will have to agree on all of this.'
Note that Clinton says 'I don't think the Europeans would like this very much.' Does he mean the Europeans would not like Russia providing security for them with a missile shield? Of course not, that wouldn't make any sense.
He means the Europeans would not like being absorbed into Russia's sphere of influence. They know what happened to the Soviet vassal states, and they don't want a return to those years.
There is no way to read the words 'Just give Europe to Russia' and 'I will take Europe' as 'I just want a missile defence system, it's no big deal.' The same applies to his intentions for Asia. Clinton understands this perfectly well, as his reply shows.
The broader context of the discussion is Yeltsin's concerns about foreign intervention in his war on Chechnya, which he sees as a uniquely European issue that non-Europeans should stay out of. This is such a critical issue for Yeltsin that he raises it immediately at the start of their meeting:
This is how the National Security Archive describes the entire exchange:
So it's not difficult to understand why this memo was classified as 'Secret', because its contents is politically explosive. You can't hand wave this and pretend Yeltsin meant something entirely different.