信用 and 確認: someone's trust versus a specific check
Compare trust gained or lost through promises and lies with checking a ticket and wallet or a reservation confirmation email that has not yet arrived.
Trust earned through keeping promises
信用 (しんよう) refers here to someone's trust. “約束を守ることで相手の信用を得られる。” says that keeping promises can earn the other person's trust.
“何度も嘘をついたため、彼は信用を失った。” reports that he lost trust because he lied repeatedly. Distinguish the first example's possibility of earning trust from the second's actual loss.
Checked belongings and an email not yet received
確認 (かくにん) means a check or confirmation. “出発前に切符と財布を確認した。” describes checking a ticket and wallet before leaving. The two belongings are the specific objects checked.
“予約の確認メールがまだ届いていない。” says that the reservation confirmation email has not yet arrived. It is the email that is missing so far; the sentence does not state that no reservation exists.
A relationship of trust versus a checking procedure
The 信用 examples concern earning or losing someone's trust. The 確認 examples concern checking belongings or an email about a particular reservation.
Do not add a completed reservation check merely because someone is trusted. An email not yet received also does not establish lying or lost trust; keep its arrival status separate.
Does an absent reservation confirmation email prove there is no reservation or that someone lied?
Show the answer and why
No. The sentence states only that the confirmation email has not yet arrived. It does not establish a missing reservation or a lie; 信用 and a specific 確認 concern different things.