promptly, namely
卽 centers on immediacy and direct connection: something happening promptly or being named as exactly that thing. The two given meanings both express a sense of 'without delay or distance'.
卽 is the older form of modern 即. It combines 皀 (a food vessel) with 卩 (a kneeling person), originally depicting a person about to eat, which extended to the idea of immediacy or 'about to'. The modern simplified form 即 replaced the left side with a more abstract shape.
The left side 皀 looks like a container of food, and the right side 卩 is a person kneeling. Imagine someone kneeling right in front of a meal, about to eat promptly — that's the immediacy of 卽.
For ソク, picture a sock puppet named Soku who always acts promptly: as soon as you say 'Soku!', it pops up immediately. Sock -> ソク, and Soku's promptness matches the kanji meaning.