- What semantic notion connects the bolded meaning beneath with all the others that aren't related to recompense? To me, nouns like remittance or solatium (if we prefer an uncommon term) fit the bolded meaning more intuitively.
mid-14c., consideracioun, "a beholding, looking at," also "a keeping in mind," also "contemplation, reflection,"
from Old French consideracion (12c., Modern French considération) and directly from Latin considerationem (nominative consideratio) "consideration, contemplation, reflection," noun of action from past-participle stem of considerare "to look at closely, observe" (see consider).
Meaning "a taking into account, act of paying attention to" is from late 14c.; that of "examination, observation" is from early 15c.. Sense of "thoughtful or sympathetic regard" is from c. 1400. Meaning "that which is or should be considered" is from late 15c. Meaning "something given in payment" (as recompense for service) is from c. 1600.
- Which semantic shift is this?
The bolded meaning looks related to meaning of 'consideration' in English contract law. See Paul Richards, Law of Contract (13 edn 2017), pp. 68 Bottom-69:
[...] it has become preferable today to think in terms of consideration amounting to a claimant buying a defendant’s promise by performing some act in return for it. Alternatively, the claimant may purchase the defendant’s promise by the furnishing of a counter-promise. This modern approach was summed up by Sir Frederick Pollock (1950) in Principles of Contract, where he defined consideration as:
An act or forbearance of one party, or the promise thereof, is the price for which the promise of the other is bought, and the promise thus given for value is enforceable.
This definition was approved by the House of Lords in Dunlop Pneumatic Tyre Co. Ltd v Selfridge and Co. Ltd  AC 847 and is regarded as being more representative of the doctrine of consideration in the modern commercial contract than the nineteenth-century concept of benefit and detriment.