Bottom line up front:

• Items that do not normally require UMD to activate (e.g. most use activated, command word, or continuous items) don’t suddenly start needing UMD just because an artificer made them.
• For items that do usually require UMD (e.g. spell trigger and spell completion items):
  • Anyone can activate an artificer-made item with UMD.
  • Infusions do not count as spells for the purpose of activating these items without UMD, even if they have the same name/effect. They don’t count even if the item itself holds an infusion rather than a spell (e.g. minor schemas). Thus, (single-class) artificers must always roll UMD.
  • Spellcasters who have the spell contained in an artificer-made magic item:
    • can activate all spell trigger items (e.g. wands, staves) without UMD
    • can activate some spell completion items (e.g. minor schemas) without UMD
    • cannot activate other spell completion items (e.g. scrolls) without UMD

Now for details on each case: Can the item be activated without UMD?

Item mimicking an infusion you have: No

First of all, infusions don’t come into this. The artificer doesn’t use his infusions to create magic items, which is why he always needs the UMD check to make them and cannot make them without one even if he has an infusion with the same name/effect as the spell the item requires.

It’s also why having that infusion doesn’t allow him to activate the item without UMD.

In short, both the creation and activation of spell completion or spell trigger items (e.g. scrolls or wands) requires that you have the spell, and infusions aren’t, so the artificer must resort to UMD for all cases.

This fact says nothing about what’s going on for those who do have the spell as an actual spell. For them…

Item mimicking a spell you have: Sometimes

Spell Trigger items: Yes

Spell trigger items, like wands, very much do work. There’s only one requirement for these items, and it’s pretty easy:

Anyone with a spell on his or her spell list knows how to use a spell trigger item that stores that spell. (This is the case even for a character who can’t actually cast spells, such as a 3rd-level paladin.)

I’ll repeat the note that artificers do not have the spell on their spell list, even if they have an infusion with the same name/effect, but other people can and do.

Artificer-made items are special in that they are neither arcane nor divine (see the errata to Eberron Campaign Setting), just a unique typeless case just for artificers. However, this doesn’t matter for spell trigger items, because no one cares what type they are. Just as a cleric could use a wand of cure light wounds made by a bard, he could also use a wand of shield of faith made by an artificer.

Spell Completion items in general: Sometimes

Unlike the rules for spell trigger items, which apply to all spell trigger items, each type of spell completion item has different rules, with only one point in common. That point is this:

To use a spell completion item safely, a character must be of high enough level in the right class to cast the spell already. If he can’t already cast the spell, there’s a chance he’ll make a mistake.

This is all the game has on spell completion in general. Note that it’s not actually enough to run on: what chance? what does a mistake mean? The answers to these questions are left up to the individual types of spell completion item.

Scrolls: No

Scrolls have way more rules. Specifically, they have these rules for activating:

To have any chance of activating a scroll spell, the scroll user must meet the following requirements.

• The spell must be of the correct type (arcane or divine). Arcane spellcasters (wizards, sorcerers, and bards) can only use scrolls containing arcane spells, and divine spellcasters (clerics, druids, paladins, and rangers) can only use scrolls containing divine spells. (The type of scroll a character creates is also determined by his or her class.)

• The user must have the spell on his or her class list.

• The user must have the requisite ability score.

If the user meets all the requirements noted above, and her caster level is at least equal to the spell’s caster level, she can automatically activate the spell without a check.

Here, the scrolls that the artificer makes fail that first requirement: they are not arcane or divine, they are a special case just for artificers (see the errata to Eberron Campaign Setting). This means that no one who has the spell on their spell list is going to “be of the correct type,” as the type is unique to artificer scrolls (even the artificer himself isn’t the correct type! he’s not even a spellcaster in the first place). So, just like a cleric could not activate a scroll of cure light wounds if that had been scribed by a bard, he also could not activate a scroll of shield of faith that was scribed by an artificer.

It also means that archivists and wizards cannot scribe artificer-made scrolls into their prayerbooks and spellbooks, respectively, since the former need divine spells and the latter need arcane spells. This is, in fact, why the errata that changed artificer items to typeless was made.

Minor Schemas: Yes

Minor schemas from Magic of Eberron are spell completion items that are not scrolls. They do not use any of the scroll’s specific rules, and instead have separate rules for what happens when you attempt to use a minor schema of a spell (or infusion!) too high in level.

Importantly, minor schemas also have different rules about activation, specifically with respect to type:

Schemas have no arcane or divine designation; they are usable by any character with the spell on his spell list regardless of the type of spell he casts.

That means that artificer-made minor schemas are not unique in being typeless, but rather the norm, and doesn’t mess anything up.

Do note, however, that

As with other spell completion items, artificers must use a Use Magic Device check to use minor schemas, even if a schema’s spell or infusion appears on their class list.

So even though you can put an infusion into a minor schema, minor schemas still only care if you have the spell on your spell list, and since the artificer hasn’t got those, he’ll always have to roll UMD.

Some weirdness from Rules Compendium

Rules Compendium contains this:

Activating a Magic Item

[…]

Spell Completion

This is the activation method for scrolls. A scroll is a spell or collection of spells, that has been stored, mostly finished, in written form. All that’s left to do is perform the finishing parts of the spellcasting. Using a scroll properly involves several steps and conditions.

Decipher: The writing on a scroll must be deciphered […]

You see what’s going on here? Rules Compendium takes the rules for scrolls, verbatim as far as I can tell, and shoves them under the Spell Completion banner. It doesn’t even change those rules to say “spell completion items” rather than “scrolls” here! Ultimately, even super-strict RAW, though, this doesn’t actually change anything. Rules Compendium still doesn’t say all the rules of scrolls apply to all spell completion items (and good thing, too, since that would break minor schemas), it just sort of makes it seem like that’s what it’s doing by labeling the section Spell Completion and giving that first sentence about it. But from there it immediately goes off describing scrolls, specifically, and not spell completion items, in general.

Use Scrolls of Wish or Limited Wish

The Wish scrolls cost 28,825 gp and allow you to:

Duplicate any wizard or sorcerer spell of 8th level or lower, provided the spell is not of a school prohibited to you.

Duplicate any other spell of 6th level or lower, provided the spell is not of a school prohibited to you.

Duplicate any wizard or sorcerer spell of 7th level or lower even if it’s of a prohibited school.

Duplicate any other spell of 5th level or lower even if it’s of a prohibited school.

The Limited Wish scrolls cost 3,775 gp and allow you to:

Duplicate any sorcerer/wizard spell of 6th level or lower, provided the spell is not of a school prohibited to you.

Duplicate any other spell of 5th level or lower, provided the spell is not of a school prohibited to you.

Duplicate any sorcerer/wizard spell of 5th level or lower, even if it’s of a prohibited school.

Duplicate any other spell of 4th level or lower, even if it’s of a prohibited school.

It's not "the best" answer, but you can cast any of these things at your own Caster Level (probably 20) and allow you to cast any spell up to 6th level, assuming you are a Generalist.

(You could also just them as a spell, but then they costs XP to use)