The Invitatory must be used before the Office of Readings if that is the first Office of the day. If Morning Prayer (Lauds) is the first Office of the day the Invitatory may be used; it is optional. If any other Office (Prayer during the Day, Evening Prayer, Night Prayer) is your first Office of the day you do not used the Invitatory. Cf. The General Instruction on the Liturgy of the Hours #35.
I am going to seem like I am splitting hairs here, but the Invitatory consists of two parts: the verse “Lord open my lips…”, and the invitatory psalm with its antiphon.
The invitatory is always said in full (verse, antiphon and psalm), if it precedes the Office of Readings (or Vigils in communities that still celebrate that Office).
If the first prayer of the day is Lauds (morning prayer) the invitatory is always said, but the
psalm and antiphon are not mandatory. That is, you would use the first part of the invitatory (“Lord open my lips…”) but you
may at your discretion omit the psalm and its antiphon.
But you are still saying the invitatory, just an abridged version.
So to summarize: if the Office of Readings is first, you say the invitatory with its psalm. If Lauds is first, you say the invitatory, with or without the invitatory psalm. And if you miss
both Lauds
and the OOR, you never say the invitatory.
Here are the relevant sections from the General Instructions (my bolding):
- The whole office begins as a rule with an invitatory. This consists in the verse, Lord, open my lips. And my mouth will proclaim your praise, and Ps 95. This psalm invites the faithful each day to sing God’s praise and to listen to his voice and draws them to hope for “the Lord’s rest.” [1]
In place of Ps 95, Ps 100, Ps 67, or Ps 24 may be used as circumstances may suggest.
It is preferable to recite the invitatory psalm responsorially as it is set out in the text, that is, with the antiphon recited at the beginning, then repeated, and repeated again after each strophe.
- The invitatory is placed at the beginning of the whole sequence of the day’s prayer, that is, it precedes either morning prayer or the office of readings, whichever of these liturgical rites begins the day. The invitatory psalm with its antiphon may be omitted, however, when the invitatory is the prelude to morning prayer.
- The variation of the invitatory antiphon, to suit the different liturgical days, is indicated at its place of occurrence.
Hope this clears up the confusion and misunderstandings.