TL;DR - ask someone who has survived 45 years after treatment of their testicular cancer. The patient (and most doctors) would consider that a cure.
I believe the kxcd cartoon explains prognosis quite well. There is something that @Crags has overlooked, though, and that is the caveat "Once most cancers spread out into your body, they're incurable".
I would consider that sentence to be quite correct. A good prognosis is very different from a cure. Most of the surgeons, radiation therapists and oncologists I have worked with make that distinction. (Even surgeons, whose mottoes include "A chance to cut is a chance to cure." and "Nothing heals like cold hard steel." are careful not to use the word cure, unless it is a cure.)
I am not an oncologist, but oncologists do write papers. From one of them:
METHODS: We analysed postmenopausal women with localised invasive breast cancer. The primary endpoint disease-free survival (DFS), and the secondary endpoints time to recurrence (TTR), incidence of new contralateral breast cancer (CLBC), time to distant recurrence (TTDR), overall survival (OS), and death after recurrence were assessed... FINDINGS: At a median follow-up of 100 months (range 0-126), DFS, TTR, TTDR, and CLBC were improved significantly in the ITT and hormone-receptor-positive populations.
Although this study only followed patients for 100 months (8.3 years), some studies are much longer, and a cure is what you or I would think of as a cure: disease-free survival, not of months, but for a lifetime, that is, non-recurrence of the cancer.
Your professors sound like they are using the standard approach to cure, which is to hedge one's bets:
In medicine, a disease is considered cured when it’s been successfully treated and does not return. The concept of “cure” is difficult to apply to cancer because undetected cancer cells can sometimes remain in the body after treatment, causing the cancer to return later (referred to as a recurrence or relapse). Many cancers are considered “cured” when there is no cancer detected five years after diagnosis. However, recurrence after five years is still possible.
In the past, cure was defined as 5-years cancer free. We now know that that is survival, not cure. The same can be said of 10 years (thought that one runs more true to cure). The fact is, most studies don't follow patients indefinitely. That doesn't mean no cancer is "cured". The patients (even many of those in the xkcd comic) without recurrence are cured. Therefore it is a fallacy to say cancer can never be cured. Semantics and statistics are at play.
As I said in my comments, there are curable cancers.
Curable cancers: non-invasive basal cell carcinoma:
Nearly every basal cell cancer can be cured, especially when the cancer is found early and treated.
(This is not a hedge. The authors are lumping aggressive BCCs in here with the less aggressive BCCs.)
Certain early prostatic cancers:
These results suggest that a patient has a high likelihood of biochemical cure after treatment for prostate carcinoma with conventional doses of external beam RT if he has not demonstrated biochemical failure within 5 years of treatment. Patients with lower pretreatment PSA levels and lower Gleason scores may require longer follow-up than those with less favorable characteristics to achieve the same certainty of cure. Patients who achieve a PSA nadir ≤ 0.4 ng/mL and require ≥ 2.0 years to reach this nadir have the highest probability of cure.
Early cervical cancer can be cured by removing or destroying the precancerous or cancerous tissue.
Conclusion: Patients who survive 10 years appear to be cured of their disease, whereas approximately one third of actual 5-year survivors succumb to a cancer-related death. In well-selected patients, there is at least a one in six chance of cure after hepatectomy for CLM. The presence of poor prognostic factors does not preclude the possibility of long-term survival and cure.
Childhood acute lymphocytic leukemia:
Impressive improvements of survival rates in pediatric acute lymphoblastic leukemia (ALL) have been achieved during the last decades.1–21 Today, a long-term cure can be attained for approximately 75% of patients.
(Although good survival rates - defined by *event free survival - are defined at 6 years in this paper, there are papers which follow to 15 years with little change in the group. Like most studies, longer times mean more patients are "lost to follow-up".)
It goes on and on.
Finally, my comment was an admonition to you to be wise with your words, not an absolute.
Understanding Statistics Used to Guide Prognosis and Evaluate Treatment
The cure of cancer: A European perspective
Basal cell carcinoma: Diagnosis, treatment, and outcome
Risk-adjusted therapy of acute lymphoblastic leukemia can decrease treatment burden and improve survival: treatment results of 2169 unselected pediatric and adolescent patients enrolled in the trial ALL-BFM 95