A Heegaard splitting of a 3-manifold $ M = H \cup H' $ is stabilised if there exist essential discs in $ H $ and $ H' $ whose boundary curves intersect transversely at a single point. Equivalently, a genus $ g $ Heegaard splitting is stabilised if it is obtained from a lower genus Heegaard splitting by adding an unknotted handle to each handlebody (more formally, see the bottom of page 1 of this paper).
Usually when I see the word 'stable' in mathematics, it refers to a process which terminates: that is, one takes an object, performs some 'stablisation' procedure a certain number of times until the procedure becomes stationary, and then stops: so a stable object is one which is equal (or isomorphic or isotopic or whatever) to its stablisation, or something similar. Clearly this is not the case here, since you can stablise any Heegaard splitting and get one of higher genus, even one which is already stable.
Question(s). What is the etymology of the term "stabilised" in reference to Heegard splittings? Is there a historical reference (e.g. the first author to use this term) which gives some justification for this choice of word?
My guess is that it has something to do with the stabilisation conjecture, which was that a single stabilisation always sufficed to turn non-equivalent Heegaard splittings into equivalent ones: if a Heegaard splitting was stable, according to the conjecture, it was unique (so the 'terminating procedure' here is going up the chain of Heegaard splittings until you reach the first one where everything is unique, and then uniqueness as a property is preserved if you keep going). Therefore, an answer to my question could be a citation which proves that this is the historic reason why the term was chosen---even implicitly: if the first known use of the word "stabilisation" for certain Heegaard splittings is in conjunction with this conjecture or related theorems then that is an answer.