1. Question Restatement:
In the narrative of Exodus 4-14 (spanning from the time of Moses's initial assignment until the crossing of the Red Sea), there are three different verbs used to describe ... Pharaoh's heart, generally all translated in English versions as "harden" or hardened": חזק,כבד,קשׁה
I have failed to find a pattern. Is there any distinction intended between these? Or why use three different terms?
NASB, Ex. 7:3 - But I will [incite, אַקְשֶׁ֖ה] Pharaoh’s [לֵ֣ב, heart/mind] that I may multiply My signs and My wonders in the land of Egypt.
2. Clarifications Regarding The Question:
Just a few clarifications so the answer makes more sense!
In Hebrew Scripture, (even Greek Classics), "Heart" is in reference to the most inward part, (i.e., your thoughts), and "Spirit" is in reference to emotions, (Abraham said in his heart, Gen. 17:17) ...
NASB, Deut. 2:30 - But Sihon king of Heshbon was not willing for us to pass through his land; for the Lord your God hardened [incited, הִקְשָׁה֩] his spirit ר֫וּחַ and made his heart לֵבָב - obstinate, [Determined, Strengthened, וְאִמֵּץ֙], in order to deliver him into your hand, as he is today.
כָּבֵד - Always connotes "burden", a sense of being "overwhelmed".
חָזַק along with אָמַץ - When used as an attribute of the heart, (the mind), very nearly mean the same thing. אָמַץ is also another, very important, term used in this "context", though not Pharaoh specifically.
קָשָׁה and/or קָשֶׁה - can always be understood to mean "incited", or "Contentious", "Severely Against".
In Scripture, "קשׁה" against God is bad - but Having a "Strong Heart" is not Necessarily Bad:
NASB, Is. 35:3-5 - Encourage the exhausted, and strengthen [חַזְּק֖וּ] the feeble. 4 Say to those with anxious heart, “Take courage [חִזְק֖וּ], fear not. Behold, your God will come with vengeance; The recompense of God will come, But He will save you.” 5 Then the eyes of the blind will be opened And the ears of the deaf will be unstopped.
3. Answer - God Said He Would, But was Pharaoh's Heart actually Ever "קשׁה / Incited" against God?
NASB, Exodus 7:3 - But I will harden [אַקְשֶׁ֖ה] Pharaoh's heart.
Even before the first plague, Pharaoh's heart was already overburdened, (כָּבֵ֖ד, Ex. 7:14), and by the end of the first plague Pharaoh's heart was strengthened/ more determined, (חֱזַ֤ק Ex. 7:22).
But if God said Pharaoh's heart would be "incited", doesn't it mean it happened?
The significance is: קשׁה is never actually attributed to Pharaoh - even after his son is killed - but it is attributed to God's own people - over, and over again, (List 1 list 2) - that's an impossible coincidence.
But if it didn't happen, then why would God tell Israel to teach their sons that Pharaoh's Heart was Incited?
NASB, Exodus 13:14 - 15 - And it shall be when your son asks you in time to come, saying, ‘What is this?’ then you shall say to him ... Pharaoh was stubborn [הִקְשָׁ֣ה] about letting us go.
In a way, it's pretty much the same exact question, "why would God deliberately cause an oppressor to stumble? -- for Justice.
NASB, Ez. 20:3 - 24 because they had not observed My ordinances, but had rejected My statutes and had profaned My sabbaths, and their eyes were on the idols of their fathers. 25 I also gave them statutes that were not good and ordinances by which they could not live; 26 and I pronounced them unclean because of their gifts, in that they caused all their firstborn to pass through the fire so that I might make them desolate, in order that they might know that I am the Lord.”’
3.1. The Text Indicates that God's Purpose was To Reconcile All Nations:
NASB, Ezekiel 16:52 - Also bear your disgrace [Israel] in that you have made judgment favorable for your sisters, (Egypt, Sodom, Gomorrah, etc).
NASB, Is. 63:17 - Why, O Lord, do You cause us to stray from Your ways And harden [תַּקְשִׁ֥יחַ] our heart from fearing You? Return for the sake of Your servants, the tribes of Your heritage. 65:1 “I permitted Myself to be sought by those who did not ask for Me; I permitted Myself to be found by those who did not seek Me. ... To a nation which did not call on My name.
And the Earliest Christian / Jewish Affirmation:
Romans 11:15 - For if their rejection is the reconciliation of the world, what will their acceptance be but life from the dead?