Does the "rarity" variation mean that the DM picks the +1,+2 or +3 bonus and it just stays that way for the rest of the campaign? Or is there something else that determines the bonus in different situations?

I think there might be a typo on the source, here!  I don't believe the basic rules have 212 pages, but I did find it in the DMG on page 212.

Quote from LicoMaven >>

Does the "rarity" variation mean that the DM picks the +1,+2 or +3 bonus and it just stays that way for the rest of the campaign? Or is there something else that determines the bonus in different situations?

 It means that there are three versions, where the +1 version is uncommon, +2 rare und +3 very rare. All or any of them can exist if the DM decides they do. If the DM allows for magic item vendors, the general guidelines can be used to determine the value of these items: 500gp for +1, 5000 for +2 and 50000 for +3. Again, of course all of this is a guideline.

Equipping the item doesn't seem to add the bonus to the character sheet. What is the best way to add the attack bonus?

Just to make sure, this is an attunement item so it must be both equipped and attuned for the bonus to be accounted for.

There seems to be a bug in this item. If you make a homebrew copy, you can see that it gives a bonus to "spell attacks", but "ranged spell attacks" is a separate bonus that was (I assume) forgotten.

Make a copy and add the property, then enable homebrew content for you character and you should be able to add it.

Its basically like a +1 weapon for spellcasters spell attacks, so the rarity determines the + bonus.

So, this would NOT stack with the Artificer's empoowered wand infusion right?  

Quote from Zwordsman >>

So, this would NOT stack with the Artificer's empowered wand infusion right?  

 This being fifth edition its safe to assume no stacking unless it explicity says otherwise.  I do wonder, though, if a different item was infused if you could get any stacking spell attack bonuses... likely not.  

All versions of this item are bugged.

I cannot create any item with Spell Attack or Ranged Spell Attack that actually works on the character page currently.

by the way it says "  requires attunement by a spellcaster " i assume that it can be used by non wand wielding classes like cleiric and druid, so can if you are a cleric, would it be fair to flavor the magic item to instead of being a wand, being a holy symbol blessed by your god. 

Can anybody give me an estimated cost on the 3 variations 

of the wand? 

I can. Roughly 50,000 GP for a very rare, which is the +3 version. I suggest as a player doing a quest to defray cost or making one yourself. As a DM you should offer quests for it. 

Artificer Infusion cannot be put on Magic Items, but if you mean if you have both of them on different hands, then yes, they would stack, Magic Items of the same Name do not stack, but if they are different things then they should stack

Not to necro a year old post, but it actually should. Both items state that you only need to hold and be attuned to them for the bonus.

Except that each item is a unique spellcasting focus which grants a +1 bonus to spell attacks/etc. You cannot use more than one focus at a time when casting spells, so you can either choose the +1 want, or the +1 artificer infusion each time you cast a spell. Likewise if you had a +2 wand and a +1 enhanced focus, you wouldn't get a cumulative +3, you would most likely choose to apply the wand's +2 bonus instead. Check the Spellcasting & Spellcasting Focuses section(s) of PHB

It doesn't state that you have to use either as a focus to gain the bonus, just that you have to hold them.  As an artificer you need to use one as a focus since all your spells have a material component, but both items say "When holding" not "when using as a spellcasting focus". This makes sense because if you had to use them as as focus for the spells to gain a bonus, classes outside of artificer wouldn't be able to use them on spells without material components and no one would on spells with a consumed material component.

the DM picks how rare the individual item you found is. other versions of that item may have a different rarity

It's 5pe.

They oversimplified in the name of 'its so simple'.

Your DM can reasonably treat those two buffs as essentially identical.

You can have one but not stack two.

The argument for using both the wand of the War Mage and the enhanced Arcane Focus... Would rely on the fact that Sage advice allows Magic ranged weapons to stack with magic ammunition. 

Ironically, this is a 5e departure from previous 3.5 rulings. So while there can be debate about the rule as written, it's not easy to debate which philosophy is right or wrong.

The other intriguing problem is that magic weapons don't stack above a +3, supposedly because bounded...unless they have ammunition, in which case, they can have +3/+3 =+6 magic item enhancement...

Factor in ranged weapon archer fighting style, for +2 to hit on top... And Sharpshooter to ignore cover modifiers.

And ranged attacks apparently, can enjoy 5 points more to hit ignoring cover, than melee weapons.

Why? Reasons.... Apparently... 

https://grammar.yourdictionary.com/punctuation/when-to-use-ellipses.html