The caption from Exodus above contains neither Hebrew word related to pray. "Pray" stands as a respectful request added in by the English translators. Moses, more properly, demanded God blot him out with Israel if a death sentence fell on them. Like a proto-Christ he confirmed to God that he would forfeit his life for all of Israel rather than see them destroyed.
Professor Stronge explains palal h6419 in this manner:
to intervene, interpose, pray
(Piel) to mediate, judge
Things here get a bit circular as meaning of "pray" takes definition from the same word "pray". Therefore, we look to the other related words: "to mediate, judge, interpose, to intervene."
Examples of palal in the texts follow below:
1Sa 2:25 If one man sin against another, the judge shall judge H6419 him: but if a man sin against the LORD, who shall intreat H6419 for him? Notwithstanding they hearkened not unto the voice of their father, because the LORD would slay them.
With the lens of intercession, we see a man in authority over other people making requests from God concerning those people:
Gen 20:17 Abraham, the friend of God, makes requests to have the damages against him forgiven by God. Abimelech, who God called a dead man, had no standing to request anything from the Most High; however, God showed he his willingness to threaten to slay Abimelech for sins against God's prophet. Without a prophet / friend of God nearby, no intercession can take place.
Num 11:2 The people demanded Moses stand between Israel and God (Deu 5:4-5). The people refused to take responsibility for themselves and, against his will and God's, turned Moses into a quasi-king. Since they made him chief, only he had standing to address God directly. When Israel breaks covenant only Moses can ask for forgiveness.
Deu 9:20 Even Aaron, who God would later choose as High Priest (intecessor for Israel), had no hope against God's justice without Moses to pray. Remember this happened in accordance with Aaron's own choice as one of the people who feared God and his fire on the mountain. Unfortunately for people who do not accept direct communication with God, God still holds them accountable if they have entered into a covenant with him. Thus their transgressions, without an intercessor, can find no forgiveness and they must endure the punishment.
1Samuel 2:25 Here Samuel has given his sons the office of judges, but finds them to be completely corrupt in their handling of this office. Instead of standing in the gap between God and Israel and leading the people with right rulings, they accept bribes and favor the powerful. He chastises them and tells them they must submit to the will of God, but they ignore him because they know if they do they will be liable for the punishment. Instead they let Samuel continue as intercessor.
1Samuel 12:23 Here Samuel gives us an object lesson about his role as judge over Israel. He points out that God requires him to pray for Israel, but that true prayer is to teach the right rulings of God (aka The Torah). Samuel the Prophet has turned his will over to God completely, and the will of God is that the people learn the Torah.
From the examples above, we see that whoever attempts to palal must carry a heavy burden and stand alone for all of Israel. The assembled people turned away from their duty to communicate directly with God and instead left the task to others. With this Hebrew understanding of the one who prays let us reread these famous passages from the New Testament:
1Ti 2:1 I exhort therefore, that, first of all, supplications, prayers, intercessions, and giving of thanks, be made for all men; 2:2 For kings, and for all that are in authority; that we may lead a quiet and peaceable life in all godliness and honesty. 2:3 For this is good and acceptable in the sight of God our Saviour; 2:4 Who will have all men to be saved, and to come unto the knowledge of the truth. 2:5 For there is one God, and one mediator between God and men, the man Christ Jesus;
1Pe 2:9 But ye are a chosen generation, a royal priesthood, an holy nation, a peculiar people; that ye should shew forth the praises of him who hath called you out of darkness into his marvellous light: 2:10 Which in time past were not a people, but are now the people of God: which had not obtained mercy, but now have obtained mercy. 2:11 Dearly beloved, I beseech you as strangers and pilgrims, abstain from fleshly lusts, which war against the soul; 2:12 Having your conversation honest among the Gentiles: that, whereas they speak against you as evildoers, they may by your good works, which they shall behold, glorify God in the day of visitation. 2:13 Submit yourselves to every ordinance of man for the Lord's sake: whether it be to the king, as supreme; 2:14 Or unto governors, as unto them that are sent by him for the punishment of evildoers, and for the praise of them that do well.
Timothy and Peter share some opinions about the growing return to Torah, the in-gathering prophesied by Ezekial shown through the people from different places and backgrounds who have decided to follow Jesus, and the duty of the people of Israel as God meant for them from the beginning: kings and priests. Timothy refers to those who live within the Law of God as kings and people in authority. Living in this manner provides a life acceptable to the living God and recognizes the only mediator humanity has with God: the living Torah Yeshua (Word clothed in flesh Jesus).
Peter reiterates that Israel should stand as a royal priesthood. He points out that for so long they had not existed as God's people because they put everyone from Moses to the chronicled kings between themselves and God. At one time they lived in the land as his people, at least in name, but they divided themselves into two nations which eventually God spread throughout the Diaspora. However, at the crucifixion, the veil tore signaling no more division between God and his people. Where before someone had to intercede for them so they could receive mercy (Exodus 32:31) now the individual in-gathered seed of Abraham could seek God directly. Peter reminded them to stay away from sin, which is by definition transgression of the Torah1. He reminds the believers to do Torah ordained works before the gentiles just as if at visitation day (Yom Kippur). He admonishes them to follow the laws of man which parallel the Torah. For where man's law duplicates the Torah, evil men will suffer judgment and righteous men receive reward.
Both men call the ancient priesthood of Melchizedek into order as intercessors for all the earth, as kingly servants to teach the Torah to the earth, and as the friend's of God.
Now to define the word tefillah h8605. The Strong's says simply, “intercession, prayer, a hymn, or sacred song.” Modern Jews wrap prayer items called tefillin around their hands and heads throughout the day and pronounce specific ritual prayers. They practice this in a halakha observance of one of the laws stated by Moses:
Deu 6:1 Now these are the commandments, the statutes, and the judgments, which the LORD your God commanded to teach you, that ye might do them in the land whither ye go to possess it: 6:2 That thou mightest fear the LORD thy God, to keep all his statutes and his commandments, which I command thee, thou, and thy son, and thy son's son, all the days of thy life; and that thy days may be prolonged.
Deu 6:8 And thou shalt bind them for a sign upon thine hand, and they shall be as frontlets between thine eyes.
Deu 11:18 Therefore shall ye lay up these my words in your heart and in your soul, and bind them for a sign upon your hand, that they may be as frontlets between your eyes.
Essentially tefillah and palal boil down to this simple concept iterated by Moses, “keep all his statues and his commandments... bind them for a sign upon thine hand (the thing you do work with) and they shall be frontlets between thine eyes (cerebral cortex where you think in a deep way). Prayer is to ponder his commandments and do his works.
With this final understanding we can fulfill the order of Paul to the Tehssalonians, “pray without ceasing.” (1 Thess 5:17)