Is he who dies in pursuit of knowledge considered a martyr, and does the hadith include both religious and worldly knowledge?

The intended meaning of knowledge here is religious knowledge. As for seeking worldly knowledge like medicine to serve one's nation, one will be rewarded for it if the act is done with the right intention, because it then becomes a communal obligation (fard kifaya) and an act of worship. Obedience is considered "in the way of Allah" because it is the path that leads to His pleasure, and most often, the phrase "in the way of Allah" refers to jihad. As for the hadith, "Whoever is overtaken by death while seeking knowledge will meet Allah, and there will be nothing between him and the prophets except the rank of prophethood," and the hadith, "If death comes to a seeker of knowledge while he is in this state, he dies as a martyr," both are weak.