Reading the body of your question, I get the impression it is very much about understanding your teacher’s comments and translating actually, so I want to help with this part, which seems not to have been covered yet. The second part of my answer treats eigentlich in some detail.
Understanding your teacher’s comments
The fundamental meaning of actually (which is close to factually) is well translated by tatsächlich.
We must pay attention to what customers are actually buying/Wir müssen beachten, was die Kunden tatsächlich kaufen
But beyond that, actually is used in several other ways. That’s where the overlap with eigentlich occurs! Consider this:
Wussten Sie eigentlich, daß er verheiratet ist?/Did you actually know that he is married?
Many such occurrences of eigentlich and similar terms are regarded as fillers and discouraged by traditional teachers, but of course they are a legitimate part of colouring speech, creating pauses etc. Still traditional teachers may be right to discourage use of fillers in the early stages of learning.
The fundamental meaning of eigentlich is proper, true, original and thus harks back to eigen (own, characteristic, peculiar). Hence the mathematical term:
uneigentliches Integral/improper integral
Note that ethical impropriety cannot be described with this term! Consider this:
Er heißt eigentlich Archibald Alec Leach, nennt sich aber Cary/He was in fact (originally) named Archibald Alec Leach, but calls himself Cary.
Eigentlich hast du recht/strictly speaking (in principle, fundamentally....) you are right
Eigentlich wollten wir in Griechenland Urlaub machen. Nun kam die Stromrechnung und wir bleiben in Kleinküttlach/Originally we meant to go on vacation in Greece. Now the electricity bill arrived and we‘re staying in Little Droppington.
Accentuating or filler meanings
In many cases eigentlich works like words that change the mood of the sentence, especially modal particles, such as denn. This is why you think of it as a Flickwort.
Wie spät ist es eigentlich ~ wie spät ist es denn/How late (what time) is it
The modal shift above is not exactly translatable, but I would argue it is close to anyway. Eigentlich can transport an accusing tone as well:
Bist du eigentlich noch bei Troste?/Are you crazy?
I can’t think of an English equivalent. Eigentlich can be used to add a certain casual/in passing tone:
Hat sie eigentlich einen Freund?/Does she have a boyfriend?
In the context of modal changes your example is most definitely not wrong. If the professor doesn’t have faith in his student and the student wants to soften her claim, we could very well witness the exchange
Hast du das verstanden?