It's probably a bit of both.

We have a tendency in the contemporary world to find it amazing that anyone could believe thing like the existence of gods with animal heads. We tend to see the only reasonable theology as one involving transcendental beings, like God/Allah/Yahweh/whatever. But this is a relatively new religious development.

Most ancient conceptions of religion focused on divine immanence. We start to see this in the Upper Paleolithic, (ca. 30,000-15,000) a development of sacred spaces (Wightman, Identity and Meaning). So a place might be considered numinous, containing spiritual energy. The place itself was spiritual; it wasn't imbued by some transcendental force. Gods are perceived much the same in the ancient world. The statue, or any physical representation, of a god was where it lived. It inhabited the material. Egyptian cosmology is no different, though we see a great amount of change over 3000 years of religious thought. The New Kingdom, which is the period you are asking about specifically, was one of religious upheaval and theological development. But it is probably a good idea to go back to early theological ideas so you can see the developments.

There are 4 main versions of theological thought in Egypt (chronologically Heliopolis, Hermopolis, Memphis and Thebes). These versions are 'national', but also involve asymmetrical emphasis derived from the political or religious ascendancy of a city. These typically attempt to synthesise existing traditions in order to maintain national solidarity.

All Egyptian creation myths involve the concept of the primeval waters (Nu or Nun), representing the chaos from which creation came. These waters had always existed; there will always exist. The cosmogony of Heliopolis, (the Pyramid Texts 2400-2300 BCE), develops the idea of a self-evolving creator. In this cosmogony, the god Atum is a seed floating in the primeval waters. Atum creates the primeval mound or benben, the first hill in creation (Pyramid Texts 600). In rising as the benben, Atum is shown to be responsible for the first creation of physical material.

After creating this material, Atum produces Shu, god of atmosphere and Tefnut, goddess of rain, by parthenogenesis. In some examples, Atum masturbates and Shu and Tefnut are born from his semen (PT 527). In others, Atum sneezes Shu and spits Tefnut (PT 600). Shu creates the void, the atmosphere inside the primeval waters (Coffin Texts 75). Within the atmosphere, Shu and Tefnut create Nut and Geb. Nut is the sky, who is held up by Shu, while Geb, the earth, lies at Shu’s feet.

After the void has been shaped, the first sunrise, the ultimate act of creation, can take place. Shu then becomes the father of the Ennead, the first nine gods. Shu takes on the attribute of “eternal recurrence” and Tefnut “eternal sameness” (CT 80). This sets the pattern of existence, with Shu representing the cycles of creation while Tefnut embodies the underlying order, or ma’at, inside the void, in opposition to the chaos of the primeval waters.

The chaos in the primeval waters was the basis of the tradition in Hermopolis, an important centre of the god Thoth. The Coffin Texts mention four aspects of the waters (CT 76). These four aspects of the primeval waters, Flood, Waters, Darkness and Chaos, were later developed over time, most likely in the New Kingdom (1550-1069 BCE), into the Ogdoad (a paired male and female, Amun and Amaunet embodying hiddenness; Huh and Hauhet were formlessness; Nun and Naunet were water; and Kuk and Kauket represented darkness). In the Hermopolitan version, these elements interacted, creating an energy burst that kick-starts creation.

The cosmogony of Memphis raises their god Ptah to a higher role. The Hymn to Ptah shows how Ptah was seen as a creator. This Ptah creates the world though his thoughts and words. He is an intellectual principle, the architect of creation (Papyrus Berlin 3048). Later, the Memphite Theology, dated to around the New Kingdom (Pinch, Egyptian Mythology), expands on how Ptah created all life. It describes the role of Ptah creating by means of thought and speech:

Through the heart and through the tongue something developed into Atum's image. And great and important is Ptah, who gave life to all the gods and their kas as well through this heart and this tongue. (BM EA 498)

Now, at the end of the New Kingdom, the capital of Egypt was Thebes. The Theban god was Amun, so naturally, there was a sharp rise in his prominence. In the Theban theology, Amun is depicted as the most important god, preceding all other aspects of development in creation. Amun is the prime mover who forms the primeval waters.

While Amun is the preeminent creator god of this mythology, it cleverly merges key creators from other versions into a whole:

All the gods are three: Amun, the Sun, and Ptah, without their seconds. His identity is hidden in Amun, his is the Sun as face, his body is Ptah. Their towns are on earth, fixed for the span of Eternal Recurrence: Thebes, Heliopolis, and Memphis, according to the pattern of Eternal Sameness. (Papyrus Leiden I 350)

Amun's role then takes a new theological step. Amun doesn't just develop himself, but is seen as a transcendent god who exists independently of what he has created (Van Dijk, Myth and Mythmaking in Ancient Egypt). Amun is styled as being inherent in all things, the first example of an Egyptian conception of divine transcendence.

So we kind of get a sense here of a transcendent god, yet the other gods exist independently and also as one principle. It may simply be different levels of understanding, or a way to incorporate older traditions into new ideas. I haven't touched on it because it was an anomaly, but we also have the monolotry of Akhentaten and Atenism around 1350 BCE, which exalted one aspect of the sun, the Aten, above all other gods. On the eve of the Late Bronze Age collapse, the theology most likely to have represented the average Egyptian would have been the Theban idea, which saw Amun as a a kind of outside force. That is not to say that people, particularly your average Egyptian, wouldn't have seen the rest of the pantheon as literal, animal headed gods that inhabited statues, but another level of understanding was also possible.

The Egyptians probably thought about these things differently according to time period, the area they lived in, and their education level.

It's kind of tough to answer that because of our contemporary perspective. Regarding what your professor said, mystery cults and representation of the single divine force is probably a later Egyptian conception, with some Hellenic influence. The religious structure in Egypt changed dramatically over time, but the earlier kingdoms definitely saw multiple, distinct beings governed under the concept of Ma'at.

My background in Egyptology is stronger than most but still pretty weak, but I would recommend "Conceptions of God in Ancient Egypt" by Erik Hornung. It's sort of a one-stop shop for all your Ancient Egyptian Religious needs. It doesn't have ALL of the information readily available, but the info in the book along with the bibliography are all you need for a solid understanding of Ancient Egypt.

Followup question: Why is the Christian God typically portrayed as an old man? And is this a similar phenomenon to the Egyptian gods?