Is repenting the same as it always has been or do we in the New Covenant repent in a new way? (Roman Catholic perspective)

Short Answer: in Roman Catholic belief the dual meaning of repentance is retained from its original sense in the Old Testament.

In Biblical Hebrew, the idea of repentance is represented by two verbs: שוב shuv (to return) and נחם nacham (to feel sorrow). In the New Testament, the word translated as 'repentance' is the Greek word μετάνοια (metanoia) ... a change of mind and change of conduct, "change of mind and heart", or, "change of consciousness".

From Hebrew Roots:

In the Greek of the New Covenant, there are also two words used which parallel the Hebrew usage.
1. METAMELOMAI - to have feeling or care, concern or regret which is akin to remorse.
2. METANOEO - to have another mind, which describes that radical change whereby a sinner turns from the idols of sin and of self to God.

Repentance then is the informing and changing of the MIND stirring and directing the EMOTIONS to urge the required change, and the action of the yielded WILL in turning away from sin and to God.

Strong's Concordance

Original Word: μετανοέω Part of Speech: Verb Transliteration: metanoeó Phonetic Spelling: (met-an-o-eh'-o) Short Definition: I repent, change my mind Definition: I repent, change my mind, change the inner man (particularly with reference to acceptance of the will of God), repent.

Some Roman Catholic teaching on Repentance

REPENTANCE: Voluntary sorrow because it offends God, for having done something wrong, together with the resolve to amend one's conduct by taking the necessary means to avoid the occasions of sin. To repent is to be sorry for sin with self-condemnation. (Etym. Latin repoenitere, to be very sorry, regret intensely.)

CCC 1431 Interior repentance is a radical reorientation of our whole life, a return, a conversion to God with all our heart, an end of sin, a turning away from evil, with repugnance toward the evil actions we have committed. At the same time it entails the desire and resolution to change one's life, with hope in God's mercy and trust in the help of his grace. This conversion of heart is accompanied by a salutary pain and sadness which the Fathers called animi cruciatus (affliction of spirit) and compunctio cordis (repentance of heart).

CCC 1490 The movement of return to God, called conversion and repentance, entails sorrow for and abhorrence of sins committed, and the firm purpose of sinning no more in the future. Conversion touches the past and the future and is nourished by hope in God's mercy.

The dual nature of the original term is retained. It is also used in the meta sense that you mention in your question, with a subtle connotation of the hope that all will turn to God, or all will return to God.

CCC 1037 God predestines no one to go to hell; for this, a willful turning away from God (a mortal sin) is necessary, and persistence in it until the end. In the Eucharistic liturgy and in the daily prayers of her faithful, the Church implores the mercy of God, who does not want "any to perish, but all to come to repentance" {a turning to God, a return to God}

The meta sense of repentance is also taught as being enabled by the Holy Spirit.

CCC 1433 Since Easter, the Holy Spirit has proved "the world wrong about sin," i.e., proved that the world has not believed in Him whom the Father has sent. But this same Spirit who brings sin to light is also the Consoler who gives the human heart grace for repentance and conversion.

The RCC uses the parable of the prodigal son as an illustration of both sin, turning away from God the Father, and repentance, a return to God the Father. The son's sincere sorrow is reflected in his blunt confession: I have sinned against you and against God.

CCC 1439 The process of conversion and repentance was described by Jesus in the parable of the prodigal son, the center of which is the merciful father: the fascination of illusory freedom, the abandonment of the father's house; the extreme misery in which the son finds himself after squandering his fortune; his deep humiliation at finding himself obliged to feed swine, and still worse, at wanting to feed on the husks the pigs ate; his reflection on all he has lost; his repentance and decision to declare himself guilty before his father; the journey back; the father's generous welcome; the father's joy - all these are characteristic of the process of conversion. the beautiful robe, the ring, and the festive banquet are symbols of that new life - pure worthy, and joyful - of anyone who returns to God and to the bosom of his family, which is the Church. Only the heart of Christ Who knows the depths of his Father's love could reveal to us the abyss of his mercy in so simple and beautiful a way.

Summary

In both the personal sense, and in the communal sense, repentance in Roman Catholic belief retains the meanings from the original Hebrew and Greek. Its importance, as well as its link to salvation, is cited on over 30 articles in the Catechism.

Coda

As noted in some of the passages of the Catechism, Catholic teaching on contrition is strongly related to repentance.

Perfect and imperfect contrition (From the Catholic Encyclopedia)

Catholic teaching distinguishes a twofold hatred of sin; one, perfect contrition, rises from the love of God Who has been grievously offended; the other, imperfect contrition, arises principally from some other motives, such as loss of heaven, fear of hell, the heinousness of sin, etc. (Council of Trent, Sess. XIV, ch. iv de Contritione).

From your comment:

This is why I have written that question and because in most modern evangelical churches the understanding of repenting is not very deep or interesting.

While no reformed theologian that I know of disagrees with how important repentance is, the Reformed positions (originally presented by Martin Luther) on contrition and forgiveness is one significant place where Catholics and Protestants disagree. (This is linked to confession, reconciliation, and gets toward "off topic" for this question, though it may be a fruitful area for another independent question.)

Necessity of contrition(Catholic Encyclopedia)

Until the time of the Reformation no theologian ever thought of denying the necessity of contrition for the forgiveness of sin. But with the coming of Luther and his doctrine of justification by faith alone the absolute necessity of contrition was excluded as by a natural consequence. {Pope} Leo X in the famous Bull "Exsurge" [Denzinger, no. 751 (635)] condemned the following Lutheran position:

• "By no means believe that you are forgiven on account of your contrition, but because of Christ's words, 'Whatsoever thou shalt loose', etc. On this account I say, that if you receive the priest's absolution, believe firmly that you are absolved, and truly absolved you will be, let the contrition be as it may." Luther could not deny that in every true conversion there was grief of soul, but he asserted that this was the result of the grace of God poured into the soul at the time of justification...)

This coda was offered in case contrition and repentance are being used as synonyms in your discussion with your Evangelical associates: the Roman Catholic view won't be the same as the Evangelical view, thanks in part to the Protestant roots of many Evangelical denominations.