Introduced in Xanathar's Guide to Everything is the Barbarian Path, Path of the Zealot, which among other things, provides a feature called "Warrior of the Gods" that makes revival of this character far easier than for other characters:

At 3rd level, your soul is marked for endless battle. If a spell, such as raise dead, has the sole effect of restoring you to life (but not undeath), the caster doesn't need material components to cast the spell on you.

—Warrior of the Gods, Xanathar's Guide to Everything, pg. 11

However, this wording is somewhat tripping me up, because the spell cited as an example, Raise Dead, doesn't have the sole effect of returning a creature to life, it also confers a "resurrection sickness"-type penalty, along with performing some minor curative effects on the body itself.

You return a dead creature you touch to life, provided that it has been dead no longer than 10 days. [...]

This spell also neutralizes any poisons and cures nonmagical diseases that affected the creature at the time it died. This spell doesn't, however, remove magical diseases, curses, or similar effects; [...]

This spell closes all mortal wounds, but it doesn't restore missing body parts. [...]

Coming back from the dead is an ordeal. The target takes a −4 penalty to all attack rolls, saving throws, and ability checks. Every time the target finishes a long rest, the penalty is reduced by 1 until it disappears.

—Raise Dead, Player's Handbook, pg. 270

So already we've established a precedent that "sole effect of returning a creature to life" has a bit of a finicky definition, because it explicitly includes a spell that has additional effects beyond bringing a creature back to life.

By my count (please correct me if I'm mistaken) there are six spells in officially published 5th Edition D&D that include the effect of restoring a creature to life. Which of these spells may validly be cast without consuming a Material Component?

• 
  Revivify, 3rd level Necromancy, PHB 272


• 
  Raise Dead, 5th level Necromancy, PHB 270


• 
  Reincarnate, 5th level Transmutation, PHB 271


• 
  Resurrection, 7th level Necromancy, PHB 272


• 
  Clone, 8th level Necromancy, PHB 222


• 
  True Resurrection, 9th level Necromancy, PHB 284

Raise Dead, Revivify, and Resurrection all qualify. True Resurrection may or may not.

I interpret the Warrior of the Gods (hereafter WoG) description as referring explicitly to the resurrection portion of the spell, and not necessarily the fact that the spell does additional things. I interpret WoG this way since Raise Dead is used as the example. Raise Dead has additional effects, but we can reason that it does qualify for the cost exemption since it's included as the example. Therefore, those additional effects must not matter to WoG, so the only thing left that must matter is the resurrection itself.

Clone does additional things, too (creating the body, in this case), but that is not what disqualifies it. Clone doesn't qualify for the cost exemption since it transfers the creature's soul into a new body, rather than actually resurrecting it. This is backed up by the answer to the question you linked in your comment.

This leaves us with 4 spells upon which we can focus:
Revivify, Reincarnate, Resurrection, and True Resurrection.

Nota Bene:
There is also one more spell that is capable of returning a creature to life: Wish. If we were concerned about material costs for Wish (which we aren't), we would treat Wish similarly to how I describe True Resurrection below. Whether or not this spell would qualify for a cost exemption would rely solely on the contents of the wish in question, and not the description of the spell, per my interpretation above. However, this ultimately doesn't matter, since Wish does not require material components to cast.

That said, let us examine the 4 remaining spells in our list:

Revivify

Revivify's description reads:

You touch a creature that has died within the last minute. That creature returns to life with 1 hit point. This spell can't return to life a creature that has died of old age, nor can it restore any missing body parts.

This is straightforward enough that we can soundly reason that this spell qualifies for the exemption.

Reincarnate

This spell comes with strings attached (emphasis mine):

You touch a dead humanoid or a piece of a dead humanoid. Provided that the creature has been dead no longer than 10 days, the spell forms a new adult body for it and then calls the soul to enter that body. If the target's soul isn't free or willing to do so, the spell fails.

The magic fashions a new body for the creature to inhabit...

This spell reads similarly enough to Clone that we can consider this spell to not qualify for the exemption. We are forming a new body here, as we did with Clone.

Resurrection

The Resurrection spell reads similarly to Raise Dead:

You touch a dead creature that has been dead for no more than a century, that didn't die of old age, and that isn't undead. If its soul is free and willing, the target returns to life with all its hit points.

This spell neutralizes any poisons and cures normal diseases afflicting the creature when it died. It doesn't, however, remove magical diseases, curses, and the like; if such effects aren't removed prior to casting the spell, they afflict the target on its return to life.

This spell closes all mortal wounds and restores any missing body parts.

Coming back from the dead is an ordeal. The target takes a −4 penalty to all attack rolls, saving throws, and ability checks. Every time the target finishes a long rest, the penalty is reduced by 1 until it disappears....

If we can reason from the WoG description that Raise Dead works as written, we can also reason that Resurrection should qualify. The important difference between the spells is the effective timeframe.

True Resurrection

True Resurrection blurs the line between Clone and Raise Dead (emphasis mine):

You touch a creature that has been dead for no longer than 200 years and that died for any reason except old age. If the creature's soul is free and willing, the creature is restored to life with all its hit points.

This spell closes all wounds, neutralizes any poison, cures all diseases, and lifts any curses affecting the creature when it died. The spell replaces damaged or missing organs and limbs. If the creature was undead, it is restored to its non-undead form.

The spell can even provide a new body if the original no longer exists, in which case you must speak the creature's name. The creature then appears in an unoccupied space you choose within 10 feet of you.

By my aforementioned interpretation of WoG, this spell will qualify for the exemption only if it is not used to create a new body. If the spell is used to create a new body, we are now in Clone territory, and thus no longer qualify for the cost exemption.

Conclusion

In summation, we can comfortably say that Raise Dead, Revivify, and Resurrection all soundly qualify for the cost exemption offered by WoG. True Resurrection may or may not qualify, depending on how the spell is used.