Note 17:

Special meaning of êm when final, in verbal forms

The grapheme êm is not always pronounced with this double nasalised diphthong, but only in some words, like têm ^©, vêm and mantêm ^©, and other verbal forms derived from verbs "ter" and "vir".
(Are there any other examples? I am not aware of one. Suggestions welcome.)
On the other hand, cases where êm gets the normal single nasalised diphthong are extremely rare: the only ones I could find, using the word list in the tool Ergane, are: têmporas (temples) and êmbolo (piston).
In the song I took the sample of the word têm ^© from, is also the word demência. Here the êm simply denotes a nasalised /e~/, which gets a circumflex accent because otherwise the Stress rules would dictate that the stress be on the i.

The difference between tem ^©, and têm ^© (absense and presence of the circumflex accent) becomes a difference between the acute and the circumflex accent in verbs that have ter as their second element: Confer mantém (he maintains) and mantêm ^©, (they maintain). Other verbs with this spelling / sound behaviour are abster, ater, conter, deter, entreter, obter, reter and suster.
These accents normally indicate tongue height of the vowel (acute: low á / mid-low é or ó; circumflex: mid-low â / mid-high ê or ô). But in this special case they distinguish the single diphthong (no accent or acute) from the double diphthong (circumflex).