"At the service of" versus "in the service of"

Solution 1:

There is a distinction here: to be "in the service of X" is a permanent condition; to be "at the service of X" is a temporary situation. The sense of obligation in the former is much stronger in the former than the latter.

To be in someone's service means you (in a sense) belong to him. Vassals are in the service of their lords; soldiers are in the service of the Army. There is absolutely no sense of choice.

By contrast, to be at someone's service means you are ready or prepared to serve him. This phrase is most commonly used (today) by one who is choosing to make himself available to another, almost always temporarily.

For example, if you were to visit a wealthy friend's mansion, the butler is in the service of your host, but will typically say "I'm at your service; let me know if you need anything" to you.

It is understood that the butler's (permanent, mandatory) obligation to your host supersedes his (temporary, voluntary) obligation to you. So while he is at your service so long as you behave yourself, happily serving you tea and crumpets at your request, being in the service of his employer, he will surely remove you should you displease your host.