You're talking about the employee while the GP is talking about the employer. Both have their income (personal or corporate) taxes, so both statements are true.
I am not a tax attorney obviously, but the information I've been given (by many sources) is that this year I'll pay taxes for the state in which my employer resides, not where I've been living. I think the decider is "state in which you work", which is what I meant by "local office" (i.e. I'm still considered to "work" in NY).
New York is attempting to do that, true. Usually it is your state of residence and the state in which you physically work that tax you (both, if they are different), but New York’s novel claim is unlikely to succeed for work by non-NY residents that doesn't occur in NY.
NY’s position right now is ridiculous. I hope the courts rule that NY has no right to tax income for someone who hasn’t stepped foot in NY since March.
Usually, the place of the employees residence and the physical worksite; rules for interstate travel for work and state income tax are different per state, with some applying tax on the first day of work in the state, others having a number of days threshold.
The employer’s HQ generally has no effect, independently of it being a worksite for many employees.