Thank you for your question today, I look forward to assisting you. I have nearly 20 years of legal experience in various disciplines, including JAG.
Most employers are only going to care about the characterization of service, honorable, general, etc.
That being said, when you start applying to higher levels of employment they very well may seek out your DD Form 214.
If your discharge status says Chapter 5-17 and then the narrative description states "mental disorder" some employers are going to have a problem with that.
So, what really matters is what they choose to put in that description or narrative reason for your discharge. It really makes all the difference.
Now, as for whether or not they could give you a general discharge, technically yes. Paragraph 5.1 of the AR 635-200 specifically allows for a General (under honorable conditions) discharge for all Chapter 5 sections. However, such a characterization would require notice of the reasons for the lowered status and that paragraph makes it clear that, for Chapter 5-17 (among others), a General Discharge is typically inappropriate and an honorable should be given.