Does a time varying magnetic field produce an EMF in an open conducting loop? (slidewire generator example)?

Consider a conducting rod placed in a magnetic field. I understand that if the rod moves normally to the field, it experiences an EMF. Furthermore, if the ends of rod are connected by a wire and the rod moves such that there is a flux change,the EMF would drive a current through the loop.

1. If instead of the moving the rod to change the magnetic flux, if one were to increase the strength of the magnetic field itself would there still be an EMF generated and a current driven?

2. Does this correspond to a moving rod in a constant field or does my classroom understandng(see below) of induction only applies to flux change caused by a conductor’s motion)?**

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My classroom understanding:

When my physics class was just learning about Faraday and Lenz’s law, we were shown the example of a slidewire generator to explain how a change in magnetic flux relates to an induced EMF and a corresponding current: An open-ended rod passed perpendicularly through a constant magnetic field will see positive and negative charges gather at opposite ends, as the relative motion between the charges and the field results in a magnetic force. There is no closed loop through which to analyze flux, but there is potential between the ends of the rod as long as it is moving through the field and the positive and negative charges are being forced apart. If we move the rod again, this time sliding it along the “rails” of a c-shaped piece of wire, we now have a closed loop, sitting perpendicular to the constant magnetic field. The potential in the rod can now push a current through the loop, generating a magnetic field in the opposite direction of the constant magnetic field. The motion of the rod now not only corresponds to a magnetic force on the charges in the rod, but also a change in the area enclosed by the loop, and thus a change in magnetic flux through the loop.

This all makes perfect sense.

But there are other ways to alter the magnetic flux than just changing the area of the loop. For example, we could increase the strength of the magnetic field while holding the loop area constant, and the loop would still see an EMF, according to Faraday.

So finally to the meat of my question: If we go back to the beginning, before the rod was sitting on the rails of our loop, and instead of pushing it along through the magnetic field, we instead started changing the strength of the field, would the rod (now open-ended) still experience a separation of charges like it did when we were pushing it through a constant field?
Does the changing magnetic field around a stationary rod count as relative motion between the field and the charges in the rod, corresponding to a moving rod and a constant field? Or does my class’s explanation of induction only apply to flux caused by a changing area (physical motion of the conductor)?