As for the first tradition you posted, I believe this is in also found in Al Kafi:

abu Ja‘far (a.s.) who has said the following. “When the following verse of the Holy Quran was revealed, “On the day when We call every nation with their leaders, (Imams) . .” (17:71), people asked the Holy Prophet (s.a.), ‘Are you not the Imam of all the people altogether?’ The Holy Prophet (s.a.) said, “I am the Messenger of Allah to all the people but after me there will the Imams for the people from my family. They will rise among the people but they will be rejected. The leaders of the unbelievers and misguidance and their followers will do injustice to them. Those who support, love, follow and acknowledge their authority they are from me, with me and will meet me. Let it be known that those who would do injustice to the Imams (a.s) and reject them they are not from me and with me. I denounce them and all associations with them.”

Majlisi: صحيح
https://thaqalayn.net/hadith/1/4/25/1

Now I understand the tradition, but he doesn't seem to accept the position of "Imam of all people", but rather he replies by saying: "I am the Messenger of Allah for all people, but after me there will be Imams"

So does this seem to indicate that he is not an Imam?

by

The Prophet (s) is not negating that he is an Imam. The verse indicates that there are multiple Imams, and so the companions were confused. They said to the Prophet (s): aren’t you the Imam for everyone? They were under the impression that the Prophet (s) is the only Imam for everyone. So his response is aimed at teaching them there will be other Imams after him, but there will not be any other Messenger. He is the last universal messenger.