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In the book we have the following image for D7:

enter image description here

But every other resource I consult, all 7th chords have 4 notes (this only has 3).

What's on the book seems to match the 1st inversion for D7 (minus the A note): enter image description here

Q: Is the book just omitting this information to avoid confusing readers (since we're just starting out)?

I'm afraid that I'll have to go back and re-relearn these chords(D7 is not the only example) with the proper names/inversions

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There are two very important concepts that go into this one is more of how chords work in general and another is how chords are typically played on piano.

Because of the nature of chords and the relationship between the root and a perfect 5th, it is an extremely expendable chord tone and won't really change the quality of the chord. So leaving the 5th out will never change the chord's name so both chords are D7.

On the piano in general you'll play 3 note chords per hand on the piano. For this reason and because the 5th can be omitted this shape for 1st inversion and other positions omits the 5th in typical piano books.

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  • You CAN play 3 notes per hand. But a glance at any printed piano music, or at the hands of any classical, jazz or pop player will reveal that most of the time you don't - there are SO many other textures available! We're right about the 5th being expendable though. A triad-based chord is first defined by the root note. Is it a C, G, Eb... etc. chord? Then by the 3rd. Is it major or minor? Then it's important whether there's a 7th - one will add tension, the need to move on. The 5th is just filler. (This is a simplified explanation. Maybe over-simplified.)
    – Laurence
    Commented Dec 29, 2016 at 11:50
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    I think expendable is a very good word to convey that the omission of the p5 will not extremely alter the nature of the chord. But perhaps - "it's the most expendable" would be less debatable than "extremely" expendable. Extremely expendable implies that it is optional whereas "most expendable" implies that if you must omit a note in a chord the p5 is the most likely candidate for omission without changing what chord is perceived or implied.
    – Rockin Cowboy
    Commented Dec 30, 2016 at 22:00
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Adding to Dom's answer - in the root note of D, a strong harmonic note of A is present. That means even when an A note is omitted, its sound is still there. So, if it's not played directly, it's still heard. Often, in the more complex chords of jazz - and even the simpler ones, the first note to get dropped is the 5.

It's a good move to learn all of the inversions, with and without 5.

EDIT - it's the P5 we're discussing here - b5 and #5 are very different, not only in that they are not included in the harmonics - until they are inaubible...

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As a continuation off of the other answers, indeed, the fifth is a very expendable note for most chords. Some examples of where it can be dropped:

  • Dominant
  • Major (seventh)
  • Minor (seventh)

However, there are some chord qualities that somewhat require the fifth to actually create the quality:

  • Minor (seventh) flat five (a.k.a half-diminished)
  • Augmented
  • Diminished triad*

* = Does not necessarily apply to diminished chords with the seventh (or technically the sixth) because you can still sort-of hear that diminished sound.

In the first two chords above, the fifth is b'd and #'d respectively, so if you omit it, you completely miss that quality.

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  • 5
    To put it another way: A perfect 5th can be omitted, a diminished or augmented 5th cannot.
    – Dom ♦
    Commented Dec 28, 2016 at 21:15
  • Good point about P5. That's the harmonic that is contained within the root. b5 and #5 aren't.
    – Tim
    Commented Dec 29, 2016 at 9:48

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