No, that is wrong. OP's boss is not the data processor.
In OP's situation, her employer is the data controller. Her boss works for the data controller. Her boss is not classed as either a data controller or a data processor. If there is a data breach (as there has been), her employer is responsible as data controller regardless of whether the breach was committed by her boss or someone else working for them.
A data processor is another organisation that processes the data on behalf of the data controller. So, for example, if you bank with Nationwide, your bank statements are printed by a third party. Nationwide is the data controller, the third party that prints your bank statements is a data processor. People that work for the third party are not classed as data controllers or data processors.
If there is data relating to you and you are identifiable, directly or indirectly, from that data, that is classed as personal data and you are the data subject.
GDPR does not have the concept of a data user.
If there is data relating to an identified or identifiable living individual, GDPR applies. The only complete exemption from GDPR is for personal data processed by an individual for the purposes of their personal, family or household affairs (including recreational purposes). In all other circumstances, GDPR applies and the organisation or individual controlling the data must comply with its provisions. However, depending on who they are and what they are doing, they may be exempt from some of the provisions. So, for example, if you are a suspect in a criminal investigation, the police will store personal data relating to you, but they do not have to inform you that they are processing your data.
I've answered your Jo and Sally situation up thread. Basically, you are forwarding the email either because it is needed to fulfil a contract your employer has with Jo's employer or because your business has a legitimate interest in you forwarding the email so that Jo's query is answered. That covers the fact Jo's email address is in the email. You need an additional justification for including her medical information but, in this case, you have that because she has chosen to publish this information herself. So you are not breaching GDPR by forwarding the email.
In your Amy & Beth situation, Beth has breached GDPR by telling you this information unless she has Amy's consent or one of the other justifications applies (which, in general, they won't). You have breached GDPR by sending a text. In both cases, this means your employer has breached GDPR. If Amy chooses to take action under GDPR for compensation, she must sue her employer. She cannot sue you or Beth.
Your Amy & Beth situation is the same as OP's situation - her manager has breached GDPR by telling her team about her mental health issues.