Which characters are healers?

The game suggests that a team should have at least one healer - who are the "healers" exactly? Is everyone in the support class a healer?

Note: I'm not asking which characters can heal, I want to know which characters are labeled as "healers" by the game.

I think that the game is trying to convey the message that "healers" are useful - that is to say, it is useful for each team to have at least one character that can heal other players.

"Healer" as a Role

According to this thread, the main "healers" are: Mercy, Lucio, Ana (if the player is exceptional with her), and Zenyatta. This is also verified by the Roles page on the wiki.

"Healer" in the Common Vernacular

However, more generally, I think that any player that can heal could be classified as a "healer" in the context of the statement you read (though we would need the exact statement to verify this is the case). Nevertheless, the characters that can heal can be found on the wiki:

Ana's Biotic Rifle can heal an ally for 75 over 0.9 seconds. Biotic Grenade also heals all allies for 100 and boosts other sources of healing by 50% with an area effect of 4 meters. She can hit herself with the grenade, receiving its positive effects.

Lúcio's Crossfade, when toggled to healing mode, passively heals allies and himself within 30 meters (as long as line-of-sight is maintained) for 12.5 hit points per second, or 36 hit points per second during Amp It Up.

Mercy's Caduceus Staff can heal an ally for 60 hit points per second while they remain within 15 meters and maintains line of sight.

Soldier: 76's Biotic Field heals allies within 5 meters for 40 hit points per second. Biotic Field also heals Soldier: 76 at the same rate if he is within range.

Sombra's Hack causes Health Packs to respawn 4x faster, with healing from Hacked Health Packs being credited to Sombra as healing done.

Zenyatta's Orb of Harmony heals the targeted ally for 30 hit points per second. Transcendence heals all allies within 10 meters for 300 hit points per second (duration 6 seconds). Transcendence also heals Zenyatta in addition to making him invulnerable to damage.

All Support Characters are NOT Healers

No, not all support characters are healers. Symmetra is support, though she cannot heal. As her wiki page states:

Symmetra is the only Support hero that is not a healer.

The only healers (in alphabetical order) are:

Ana

Lucio

Mercy

Zenyatta

Those are the heroes whose main purpose/abilities/ultimate is to restore health to allied players.

This said, there are some other heroes that can provide HP support in some way. Soldier has a BIOTIC FIELD which heals anyone in small range for few seconds.

And Sombra can hack health packs to provide increased HP heal from them.

Only the four mentioned heroes above are classified as healers in the game, and it would be their main job to heal and restore health.

Symetra may be in the support section but she's not a "healer" she's a builder alongside Torbjorn. As mentioned in the comments bellow, the different classifications are very arbitrary and many heroes can fit more than one section. For instance Sombra could have been placed as a Support hero just as easily. Or Symetra could have been a defence hero or Roadhog could have been an offensive hero.