I've never heard or used the term "semi-holiday". Certainly, Red Nose Day in the UK is not regarded as any kind of holiday. It's just a day where some people wear a red nose and do silly things in order to raise money for charity, and in the evening there is a show called Comic Relief where lots of comedians and celebrities perform, again to raise money.
In the UK, a working day is either a working day or a Bank Holiday. It can't be half and half.
Student or Learner