Hunted Phantasm's color identity is blue only (and has no basic land type), so it can be included in a Commander deck whose commander's color identity includes blue, whether or not that commander's color identity also includes red (C.R. 903.5c-d).
The only things that matter for a card's color identity are—
the colored mana symbols that appear in the mana cost and rules text,
color indicators, and
characteristic-defining abilities that add colors (currently only "[This card] is all colors"),
and for a double-faced card, both its faces are considered here (C.R. 903.4, 903.4d, 207.1, 604.3). A color word such as "red" is not a colored mana symbol (compare C.R. 107.4a, 107.4e-f, colored mana symbols, with C.R. 202.2a, color words), and is currently ignored for color identity purposes due to the third point given above. Compare Hunted Phantasm with—
Door to Nothingness, which is colorless, but has all five colors among mana symbols in its rules text, so that its color identity is all five colors (C.R. 903.4), and
Sarpadian Empires, Vol. VII, which is colorless and has a colorless color identity, even though all five color words appear in its rules text.
EDIT (May 6): Added rule citation.
EDIT (Oct. 12, 2020): Edited.
EDIT (Apr. 5, 2021): One rule was renumbered in the meantime.
EDIT (Apr. 5, 2022): Moderately edited.
One more point of clarity - color identity is determined by mana symbols (WUBRG) in the card's cost and rules text, as well as a color indicator most commonly found on double face cards (for example, Archangel Avacyn is WR because Avacyn, the Purifier has the red color indicator next to the word "Legendary"). This color indicator can also be seen on the new Augment cards from Unstable. Color identity does not count words (like "red" on Hunted Phantasm) and mana symbols in reminder text (like the on Blind Obedience).
Color identity comes from mana cost (all the colors up in the cost), mana symbols in the text (for instance, Kird Chieftain is a red card, but has a red-green color identity from the green in its activated ability), and occasional weirdness (eg Transguild Courier is defined as being all 5 colors, and so can only be used in 5-color decks). Color normally is only affected by the 1st and 3rd; EDH adds mana symbols in text as a flavor rule (alongside all the other stuff with commanders and such). However, simply referring to a color (as Hunted Phantasm does) or basic land does not make it illegal.
Just for some examples:
-You can put Hunted Phantasm in an Ixidor, Reality Sculptor deck, because it would be blue in a blue deck.
-You can put Hunted Phantasm in a Jhoira of the Ghitu deck, because it would be blue in a blue/red deck.
-You cannot put Izzet Guildmage in an Ixidor, Reality Sculptor deck, because it would be blue/red in a blue deck.
-You cannot put Hunted Phantasm in a Purphoros, God of the Forge deck, because it would be blue in a red deck.
-You can't put a Swamp in a Heliod, God of the Sun deck, but oddly enough, you can add an Urborg, Tomb to Yawgmoth. Swamp has a black mana symbol in its implied "T: Add B to your mana pool" text. However, Urborg, despite looking decidedly swampy and black-mana-ey, technically only refers to the term 'Swamp', and it's okay to control Swamps once the game has started just so long as they didn't start that way in your deck.
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"It is better for all the world if, instead of waiting to execute degenerate offspring for crime or to let them starve for their imbecility, society can prevent those who are manifestly unfit from continuing their kind. The principle that sustains compulsory vaccination is broad enough to cover cutting the Fallopian tubes... Three generations of imbeciles are enough."
--Buck v Bell, 1927. This case, regarding the compulsory sterilization of inmates at mental institutions, has -- somehow -- never been overturned. Just a wee PSA for ya.
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The only things that matter for a card's color identity are—
EDIT (May 6): Added rule citation.
EDIT (Oct. 12, 2020): Edited.
EDIT (Apr. 5, 2021): One rule was renumbered in the meantime.
EDIT (Apr. 5, 2022): Moderately edited.
2023 Average Peasant Cube|and Discussion
Because I have more decks than fit in a signature
Useful Resources:
MTGSalvation tags
EDHREC
ManabaseCrafter
Just for some examples:
-You can put Hunted Phantasm in an Ixidor, Reality Sculptor deck, because it would be blue in a blue deck.
-You can put Hunted Phantasm in a Jhoira of the Ghitu deck, because it would be blue in a blue/red deck.
-You cannot put Izzet Guildmage in an Ixidor, Reality Sculptor deck, because it would be blue/red in a blue deck.
-You cannot put Hunted Phantasm in a Purphoros, God of the Forge deck, because it would be blue in a red deck.
-You can't put a Swamp in a Heliod, God of the Sun deck, but oddly enough, you can add an Urborg, Tomb to Yawgmoth. Swamp has a black mana symbol in its implied "T: Add B to your mana pool" text. However, Urborg, despite looking decidedly swampy and black-mana-ey, technically only refers to the term 'Swamp', and it's okay to control Swamps once the game has started just so long as they didn't start that way in your deck.
--Buck v Bell, 1927. This case, regarding the compulsory sterilization of inmates at mental institutions, has -- somehow -- never been overturned. Just a wee PSA for ya.