Poem is virtually synonymous with lyric.
(The Origin of the Bible: Newly Updated by F. F. Bruce, J. I. Packer, Philip W. Comfort, and Carl F. H. Henry, 2020. The Bible as Literature by Leland Ryken, Page 139)
Lyric Poem
A brief poem, often intended to be sung, that expresses the thoughts and especially the feelings of a speaker. It is identified by brief lyrics, personal or subjective voice, and a reflective or emotional mood. Unified impact is important, employed through theme and variation, analyzing how a given unit contributes to the controlling theme.
The overwhelming majority of lyrics are built upon the principle of three-part structure: an opening statement of theme (stimulus that moves the poet to sing), development of the theme, and concluding resolution. Poets develop their theme though repetition, the catalog or list, association (branching out from an initial idea to related ones), and contrast.
(The Origin of the Bible: Newly Updated by F. F. Bruce, J. I. Packer, Philip W. Comfort, and Carl F. H. Henry, 2020. The Bible as Literature by Leland Ryken, Page 139-140)
Bible
Psalms are lyric poems with a host of subgenres: lament or complaint, praise psalm.
Luke 1-2 contains nativity hymns, another kind of lyric poems. Hymn fragments are also found in the Epistles (Ephesians 5:14, 2 Timothy 2:11-13, Hebrews 1:3), and hymns of worship punctuate Revelation's visions (Revelation 4:8, 11; 5:9-10). John 1:1-18, Philippians 2:5-11, and Colossians 1:15-20 are famous Christ hymns.
(The Origin of the Bible: Newly Updated by F. F. Bruce, J. I. Packer, Philip W. Comfort, and Carl F. H. Henry, 2020. The Bible as Literature by Leland Ryken, Page 140, 142)