The gay cake case:
It is not OK to refuse to serve gay people as that's discriminatory.
It is OK to refuse to reproduce or promote a political statement you disagree with as that's against your freedom of conscious.
In the case above I bet the position would be that
a) she's made a statement which is discriminatory in a public manner who has effect on others for which she is responsible for
b) her employer found this and gave the opportunity to qualify or remove the comments, because if she is a face of their organisation this is a problem as they are associated with a comment which is problematic.
If the comment was in a private way and not associated with the organisation she would have the right to say it.
If its public and qualified - as a public interest or concern in terms of potential harm to others; for example how political correctness is preventing certain groups from being challenged for illegal behaviour that other groups would be purely because of their identity - then an organisation should uphold a right to say those things even if they disagree.
The key point being, is the intent to be prejudical to others or to raise a debate, concern or issue to another vulnerable group?
You can't just claim religious discrimination if you are just raising a belief that gay people are amoral and you think you should be allowed to say that on social media openly and in a very public way.
It's all about intent and whether your platform is public or private.
Her intent on going on social media is to promote views which are discriminatory in nature. How is someone being homosexual causing her harm?
If she is not being asked to promote views which are against her own, then there isn't an issue. She should, in theory, be able to reserve the right not to take part in pride events for example (provided the job does not explicitly state when advertised that would be her task) but the flip of that is she can't not publicly promote views which discriminate against others on the basis of identity alone either.
This is where we run into difficulties as people in the public sector for example, have a duty of care to prevent harm as part of their job. Thus if they see something which might cause harm they should be not just able to speak up but encouraged to do so. They should not fear that they will be being perceived as being discriminatory for raising concerns of harm to others who are vulnerable because one party happens to have a protected status.
The lack of understanding of purpose, intent and responsibility is where this is all falling down.