Rav Hirsch on Va'Eirah 7:3 explains the differences between the three languages of hardness.
a) קשה to be hard altogether, not to receive impressions, to let
everything pass over one without making any impression
b) כבד heavy, one can recive impressions, but there can be a big gap
between the impression and the moment one lets oneself be guided by
that impression, heavy, difficult to move, to set in motion. But, even
if it takes a lot of effort this moment can be reached,
c) חזק firm, consciously opposes any pliancy, any submission. Even the
impression is lost as far as any consequences of it go.
Thus, at first, Par'o doesn't even feel any worry or give any thought to what Moshe is saying. He totally ignores it and it passes over him like the wind. The words mean nothing to him like the sound of the wind. Once he has seen the plagues starting, he makes his will (which is symbolized by the heart) חזק. He reaffirms his will and makes a conscious effort to refuse to accept the results of what he is seeing. Then כבד, even though his conscious will might be moved, the inertia of his previous decisions are maintained so that even if he is ready to change his mind, he does not move. Note that at the final plagues (חושך and ים סוף) even when the inertia is overcome, Par'o consciously refuses to allow the pressure to move him.
Dealing with your quotes specifically, any translations will be from Rav Hirsch.
7:3: "I will 'kasheh' Paroh's heart."
I will make Par'o capable of ignoring the warnings that he will receive and of rationalizing what happens.
7:13 (after the staffs turning into snakes): "Paroh's heart was 'yechazeik'."
Par'o rationalized that his own magicians (and even the children) could do this so that he made a conscious decision to refuse the sign.
7:22 (Dam): "Paroh's heart was 'yechazeik'."
Par'o reaffirmed to himself that it was still "just magic" (since his magicians could make blood, even though they could not cancel it) and made a conscious decision to let the duel continue.
8:11 (Tzefardeia): "His heart was 'hichbed'."
Here he had to harden his heart to refuse to recognize what is going on so that he would not be affected by what he was being forced to recognize. Moshe had not only brought the frogs, but he had gotten rid of them and returned them to the river as normal, which was a difference of quality and an unpecedented precision.
8:15 (Kinnim): "Paroh's heart was 'yechazeik'."
As the third in the group that came about without warnoin (as punishment for ignoring the previous two) Par'o had to again consciously decide to reaffirm his previous decisions and deny Moshe's request.
8:28 (Arov): "Paroh 'yachbeid' his heart."
Par'o "hardened* his heart so that the effect of the Arov would not move him to recat as Moshe wanted. Again, it is a matter of making himself stubborn rather than just reaffirming an earlier decisision.
9:7 (Dever): "Paroh's heart was 'yachbeid'."
Note that "this time too" Par'o "made his heart stubborn' so that it would be difficult to move him once Hashem had showed that he could not follow through completely on his threat. After all, only those animals in the fields died.
9:12 (Shechin): "Hashem 'yechazeik' Paroh's heart."
Note that Hashem this time allowed the heart of Par'o to remain strong. That is he now gives Par'o the strength to reaffirm his earlier decision (as with Kinim) in the face of this sudden blow. As with Kinim it came with no warning as a punishment for ignoring the other two. Thus, inertia was not enough and it had to be a conscious reaffirmation of his earlier decision.
9:34-35 (Barad): "He continued sinning and 'yichbad' his heart...Paroh's heart was 'yechabeid'."
This was the first of the last group of three, and Par'o was made kaved so that it would be difficult to move him.
10:20 (Arbeh): "Hashem 'yechazeik' Paroh's heart."
10:27 (Choshech): "Hashem 'yechazeik' Paroh's heart."
11:10 (Bechoros, Krias Yam Suf - see Rashi): "Hashem 'yechazeik' Paroh's heart."
Now for the remainder, everything is based on a conscious decision to ignore the results of everything that had gone before. Had it only been the result of inertia, Par'o would have broken, the two complete sets of makos and the beginning of the third set showed that is was not lack of power that stopped Hashem from carrying through the destruction of Mtzraim, but a carefully planned and precisely executed set of operations. Thus, everything that follows had to be the result of a cold blooded calculation to see if Hashem would give up when He saw that Par'o would not break. Or perhaps a decision to see if at the very last, Hashem had to withdraw as a result of the final "sphere of influence" of the "gods" of Egypt.
Consider that the belief of the pagans was that the "gods" were a super powerful race of magicians who had their own territory and abilities and could only be defeated by a more powerful "member of their race".
In fact we see that the servants of Par'o are begging him to give in and get rid of the Bnai Yisraelin Bo 10:7 Only the will of Par'o allows them to stand.
- 13:15 (Response to Tam): "And it was when Paroh 'hiksheh' to send us..."
This last is a description of the general situation as seen from the outside. From the outside, the struggles and discussions that must have gone on, as well as the different motives, were invisible. As Rav Hirsch says
Par'o denied Hashem, refused what Hashem demanded of him, called his
own that which was through and through Hashem's. Hashem punished the
obstinacy with which the Egyptians denied this fact of belonging to
Him, by killing their firstborn.