How can the hadith be reconciled that mentions that one who sits for prayer receives half the reward of one who stands, with the hadith that states that a sick person or a traveler has recorded for him the like of what he used to do when he was healthy?

There is no contradiction between the two hadiths. The first hadith ("A man's prayer while sitting is half the prayer") discusses voluntary prayers for one who is able to stand, and it clarifies that its reward is diminished, though the prayer itself is valid. As for obligatory prayers, they are not valid unless performed standing, except in cases of incapacity. The second hadith ("If a servant falls ill or travels, what he used to do while healthy and residing [at home] is recorded for him as reward") discusses Allah's bounty upon a servant who used to consistently perform acts of obedience during his health and residency, so the reward for that deed is recorded for him if he falls ill or travels.