The two relevant meanings in the OED are A.7 (also A.6) and A.8, both in the branch II. of A. Your meaning [1.] fits with OED's A.7., and your meaning [2.], with OED's A.8. Here are the relevant entries in the OED:
II. In weakened use as an intensive or emphatic particle. (In later use many uses of senses in this branch show some suggestion of sense A. 8.)
6. Fully, completely; quite; all the way to, right to. Formerly often preceding numerals; in later use chiefly in even to (or unto), in which use some suggestion of sense A. 8 is often present. Now archaic.
1667 J. Milton Paradise Lost iii. 586
His Magnetic beam..Shoots invisible vertue even to the deep
1888 C. M. Doughty Trav. Arabia Deserta I. ii. 30
The strutting tail flowed down even to the ground.
1918 E. S. Eells Tales of Giants from Brazil ix. 151
The prince was praised throughout the kingdom and there is talk of him even unto this very day.
1997 J. Walsh Forty Martyrs Eng. & Wales 4
Martyrdom..means bearing witness even unto death.
a. Prefixed to a subject, object, or predicate, or to the expression of a qualifying circumstance, to emphasize its identity, or to reinforce the assertion being made about it; namely, that is to say, truly. Now archaic.
1954 J. R. R. Tolkien Fellowship of Ring ii. viii. 394
Maybe thou shalt find Valimar. Maybe even thou shalt find it.
a. Used to convey that what is being referred to is an extreme case in comparison with a weaker or more general one which is stated or implied in the adjacent context. Prefixed to the particular word, phrase, or clause in which the extremeness of the case is expressed.
2007 Time Out N.Y. 18 Jan. 24/3
Even the newest New Yorker knows that the furthest eastern border of Greenwich Village is Fourth Avenue.
In the comment to the meaning 8., the OED states (emphasis in bold mine),
Now the prevailing use of the word in English. The use is not attested in other Germanic languages. It is rare in regional use, and (though a natural development of A. 7) seems not to have arisen before the 16th cent., and took time to become fully established.
So, the semantic trajectory of even (in the sense that interests us here) began with it being an intensive or emphatic particle, to which gradually there became attached the connotation that the object of this emphasis is an extreme case in comparison with other similar cases (which are either stated or implied). As others have said in the comments, this sort of change is more of a progression than a shift; even the OED calls is a 'natural development' and says that it is hard to draw the line of where one ends and the other begins ('In later use many uses of senses in this branch show some suggestion of sense A. 8.'). We see one way this worked in the examples provided with A.6. One started using even in the phrases even to or even unto, which initially meant fully to, but then the implication of that being an extreme case was added. This is not particularly surprising, given that fully to implies that the thing being reached might be distant or big, i.e. extreme in some way.
It is not clear that there is a special name for this kind of etymological evolution. It doesn't sound like metonymy is relevant, given that metonymy is 'a figure of speech in which a thing or concept is referred to by the name of something closely associated with that thing or concept' (e.g. referring to the U.S. presidential staff as the White House, i.e. by the name of the building that contains it). More generally, none of the tropes listed on the Wikipedia's article 'Figure of speech' seem relevant.