Telekinesis, a level 5 spell, says you can manipulate an object. I would think that the example of opening a door is at least as complex as pushing a button on a rod.
You can exert fine control on objects with your telekinetic grip, such as manipulating a simple tool, opening a door or a container, stowing or retrieving an item from an open container, or pouring the contents from a vial.
Also it does not say that you cannot manipulate a magical object. So I think this one is valid.
Unseen Servant, a level 1 spell, should also be able to do it.
The servant can perform simple tasks that a human servant could do, such as fetching things, cleaning, mending, folding clothes, lighting fires, serving food, and pouring wine.
Again, there is nothing saying that it cannot activate a magical item. (However, it would not be able to read a scroll or drink a potion)
However, you say at a distance and the Unseen Servant needs to be able to reach that location. Some people say the Unseen Servant can hover, others say it cannot.
Ten Foot Poles could also be used to push on a button. You do not specify the distance, but there are no reasons why that would not work. Only it may take a few tries (i.e. Dexterity check on a fairly high DC such as DC 16.)
Missiles could also be thrown at it (rocks, arrows, bolts, etc.) However, that may be quite a feat to hit that tiny little button (although a giant can throw big rocks, he! he!)
Rods are given a general description on page 139 of the DMG:
A scepter or just a heavy cylinder, a magic rod is typically made of metal, wood, or bone. It's about 2 or 3 feet long, 1 inch thick, and 2 to 5 pounds.
That gives you a general idea and you as DM can narrow it down to where you'd like it to be within those bounds.
Best Answer
Yes and Yes. The cases aren't mutually exclusive.
Both the described moving the object by the caster to be fixed in it's new position and using the fixed nature of the object are available to the caster.
Caster moves the object, it still remains under effect of the spell.
Moving the object does not end the effect of the spell. So the effects still apply in that the object is fixed in place for everyone but the caster.
The caster and designees move the object at their discretion.
The use of "can" does not obligate the caster to move the object normally. This interpretation hinges on the term can. In short, the sense of "can" applicable here indicates ability to do a thing and indicating possibility to do it. The sense here is synonymous with "may".
Contrast the existing wording with that where "can" has been removed. It would essentially be "You move the object normally" indicating the spell effects do not apply to the caster. The use of "can" gives the caster the option to use the effects of the spell or move the object normally.