No, you cannot counterspell a magic item. Unless something says you can...
As you pointed out, counterspell requires someone to, "see a creature within 60 feet of you casting a spell."
So they must see a spell being cast.
In the DMG, under Activating an Item it says:
Some magic items allow the user to cast a spell from the item, often by expending charges from it. The spell is cast at the lowest possible spell and caster level, doesn’t expend any of the user’s spell slots, and requires no components unless the item’s description says otherwise. The spell uses its normal casting time, range, and duration, and the user of the item must concentrate if the spell requires concentration. Many items, such as potions, bypass the casting of a spell and confer the spell’s effects with their usual duration. Certain items make exceptions to these rules, changing the casting time, duration, or other parts of a spell.
I've bolded the important parts. There are no components (unless specified in the description) to these spells being cast:
- There is no verbal component so there is no audio clue that a spell is being cast
- There is no somatic component so there is no waving of hands or grand physical gestures1 to see a spell is being cast
- There is no material component (other than the item itself), but since most magical items don't have a standard form2, there is nothing to say this thing is casting a spell
Since nothing signifies that a magic item is casting a spell, there is nothing to see, and so nothing to counter.
Nothing in the descriptions say a magic item needs to have an indicator that it's being used. Except those that do...
There is nothing in the rule books or general item descriptions that say, "To activate a charge in a wand you need to point it and say a magic word, and press a button." Here is the description for thw wand of fireballs:
This wand has 7 charges. While holding it, you can use an action to expend 1 or more of its charges to cast the fireball spell (save DC 15) from it. For 1 charge, you cast the 3rd-level version of the spell. You can increase the spell slot level by one for each additional charge you expend.
You just "hold it" and "expend one or more charges," that's it. And considering that this item requires attunement, the holder must have some sort of bond with the device. I would put it firmly in the realm of possibility that the holder need only think to make the wand work.
Now, let's look at something like the driftglobe:
This small sphere of thick glass weighs 1 pound. If you are within 60 feet of it, you can speak its command word and cause it to emanate the light or daylight spell. Once used, the daylight effect can't be used again until the next dawn.
You can speak another command word as an action to make the illuminated globe rise into the air and float no more than 5 feet off the ground.
Here, there is a clear, perceivable action that must be taken; a spoken word. In this case, it would be DM fiat if speaking the "command word" is the same as a verbal component and thus a clue that a spell is being cast.
But I need to point out that for spells like command and suggestion the verbal command/suggestion the caster gives is NOT considered the verbal component. So there is precedence to verbal components being different. It's not a sure thing that a "command word" is part of a spell, and as such, still uncounterable.
As a side note...
It also states that some items bypass the whole casting of the spell and it just happens. Like if you drank a potion of flying.
So in these cases, there is no spell being cast that is seen and therefor no way to counterspell.
1 Well, there could be a waving of hands if that is what activates the device, but it is just as likely that a wand could be activated by a button. That's up to a DM. But unless a device calls out an action the wielder must take, it could be as simple as thinking.
2 A wand is a wand, but a wand can look like a pointy stick, a metal rod, glass tube, or a rock. It's up to the DM to decide what each one looks like so people can't say, "That looks like a Wand of Fireballs!"