Both these definitions are appropriate when the Carmelite tradition speaks about 'aspirative prayer'. One the on hand, the 'Prayer of Aspiration' is about desiring contemplative union with God. On the other, it is like practising the presence of God so fully in our lives that we are effectively alive with the breath of God.
The 'Prayer of Aspiration' is often offered as short exclamations, brief breaths, that express a desire for God.
The following text is by Dutch Carmelite Sanny Bruijns:
In the writings of Carmelite reformers like John of the Cross and John of Saint-Samson, we find the prayer of aspiration. Aspiration comes from the word 'aspirare' and this means breathing towards. This refers to inhaling the love and moving along the path that love opens up in us. It is the praying that happens in you. In a receptive soul God breathes in order to sarry the soul along with him. According to Hendrik Herp, the contemplative life consists of a soul and a body. Love is the soul of contemplation and aspiration is the body of contemplation. Through the aspiration, or rather through moving along the movement of God's breath, the desire of man to love will enkindly his love ever more. When love lives in us, then every word we speak will be like an arrow which ierces the heart of the other. Aspirations are short and quick prayers of heaving sighs. The sighing of the soul, which utters an 'oh' or an 'ah' is like the prayer of someone who does not know how to pray. It is like the Spirit, who comes to rescue us in weakness and pleads with unspeakable sighs. In his letter to the Romans, St. Paul speaks about his hope for the coming glory. With St. Paul, Carmelites live from hope. Our praying is like the coming of the Messiah. Our working in silence is carried by the hope which lives in our heart. But 'because our hope is directed towards the invisible, our waiting and expecting must be accompanied with perseverance.'
Printed in 'The Carmelite Way of Prayer', Carmel in the World magazine, Vol. LIII, No. 3 (2014), pp. 231-34 [p.234].