Partially nested/partially crossed structure in a mixed model

Hi,

I already apologize if this is a rather basic question, but I would really
appreciate some advice as I am unfamiliar with the particular design issues
here:

I am assessing cheating during exams depending on the type of exam (online
test vs. on-site test). Some of the students wrote only online exams,
others only on-site exams and some did both. We assessed cheating for both
types of exams, meaning that students who did both types of exams answered
the cheating questions twice (for each type of exam) and others who wrote
only one type answered them only once. So the data looks  like this:

Subject ExamType Cheating
1 online 2
1 on-site NA
2 online NA
2 on-site 1
3 online 4
3 on-site 3
...

I understood I am dealing with some sort of partially nested/partially
crossed fixed effect here. My question is, is it appropriate to analyze the
effect of online vs. offline testing within the same model if I just add a
random intercept for the subject? So far what I came up with would look
like this:

Cheating ~ ExamType +(1 | Subject), data = df

The model looks (too?) simple,  the model converges and the obtained
results look reasonable. But I cannot help the sense I may be overlooking
something, as of course there is a lot of data missing by design in the
dependent variable and I am not sure whether lme/lmer handles this
correctly? I would be happy for some expert to comment on this or
alternatively, any literatur advice on the topic. Thank you so much in
advance!

Best

Selma