This problem is not conceptually different from a standard multi-category 1-way ANOVA, except that it involves logistic regression instead of (the equivalent of) standard linear regression. You thus have two issues to address: whether there are any significant differences at all among the 10 cancer types with respect to the probability of seizures, and if so which cancer types differ in that respect.
A logistic regression of seizure (binary 0/1) against a single 10-category cancer-type variable addresses the first issue. Software will typically report coefficients and p-values for differences of each of 9 cancer types from a single reference cancer type, but what you first care about here is the model as a whole. If the model is not significant overall, then you simply stop and say that there are no significant differences among these cancer types.
If the model is significant overall then you can proceed with examining differences among cancer types. Absent pre-specified hypothesis you must take into account the issues raised by multiple comparisons. This page shows a way to proceed for logistic regressions.
I understand the initial appeal of testing each cancer type against the average of all the other 9, but that can lead to all sorts of problems. For example, say that 5 types of cancer had no seizures while all patients with the other 5 types had seizures. There clearly are differences among cancers with respect to seizure probability, but if the number of cases is limited you might find no individual cancer "significantly different" from the approximately 40% - 60% seizure incidence in the average of the other 9. Combined with the lack of any test of overall difference, you also face a substantial problem with multiple comparisons as it's pretty easy to get one "significant" result by chance out of 10 comparisons even if there are no actual differences (happens about 40% of the time at p < 0.05).
A joint test for any differences among cancers followed by an analysis of individual differences that controls for multiple comparisons is the tried-and-true way to approach this type of problem.