A possible way to understand the different usage is to look to the Targum Onkelos.
In the examples you cite, Shemot 14:17 and Shemot 14:25:
וַאֲנִ֗י הִנְנִ֤י מְחַזֵּק֙ אֶת־לֵ֣ב מִצְרַ֔יִם וְיָבֹ֖אוּ אַחֲרֵיהֶ֑ם וְאִכָּבְדָ֤ה בְּפַרְעֹה֙ וּבְכָל־חֵיל֔וֹ בְּרִכְבּ֖וֹ וּבְפָרָשָֽׁיו׃
וַיָּ֗סַר אֵ֚ת אֹפַ֣ן מַרְכְּבֹתָ֔יו וַֽיְנַהֲגֵ֖הוּ בִּכְבֵדֻ֑ת וַיֹּ֣אמֶר מִצְרַ֗יִם אָנ֙וּסָה֙ מִפְּנֵ֣י יִשְׂרָאֵ֔ל כִּ֣י יְהוָ֔ה נִלְחָ֥ם לָהֶ֖ם בְּמִצְרָֽיִם
The first 'מחזק' is a verb and has a targum of 'מתקף'. Jastrow gives a translation of: to overpower, or seize, or to make irascible or vehement. So in the context of effecting Pharoah's heart, it means changing his natural emotional state at the time.
The 'ואכבדה' in the same line is also a verb and has a targum of 'ואתיקר' which means according to Jastrow to make heavy. This was being done to the chariots and the horses of the cavalry.
In verse 14:25 the 'בכבדת' is a noun and has a targum of 'בתקוף' which Jastrow translates as force or power. So in context, the binding of the wheel of his chariots was done through the force (of gravity) making them heavier than normal or frictional resistance from the mud binding them from moving or possibly both.
The posuk mentioned above, Shemot 8:11 has a much different message.
וַיַּ֣רְא פַּרְעֹ֗ה כִּ֤י הָֽיְתָה֙ הָֽרְוָחָ֔ה וְהַכְבֵּד֙ אֶת־לִבּ֔וֹ וְלֹ֥א שָׁמַ֖ע אֲלֵהֶ֑ם כַּאֲשֶׁ֖ר דִּבֶּ֥ר יְהוָֽה׃
This mentions that Pharaoh saw 'הרוחה' and in consequence made himself heavy hearted (melancholy, a pensive sadness). The key to understanding this usage relates to the 'רוחה'. The Targum says that this relates to the open space at court where counsel was taken. In other words, Pharaoh took counsel about the frog situation but didn't listen to his counselors.
What was the 'frog' situation? For this it is necessary to look at Shemot 8:5-10.
וַיֹּ֣אמֶר מֹשֶׁ֣ה לְפַרְעֹה֮ הִתְפָּאֵ֣ר עָלַי֒ לְמָתַ֣י ׀ אַעְתִּ֣יר לְךָ֗ וְלַעֲבָדֶ֙יךָ֙ וּֽלְעַמְּךָ֔ לְהַכְרִית֙
הַֽצֲפַרְדְּעִ֔ים מִמְּךָ֖ וּמִבָּתֶּ֑יךָ רַ֥ק בַּיְאֹ֖ר תִּשָּׁאַֽרְנָה׃ וַיֹּ֖אמֶר לְמָחָ֑ר וַיֹּ֙אמֶר֙ כִּדְבָ֣רְךָ֔ גו״
וְסָר֣וּ הַֽצְפַרְדְּעִ֗ים מִמְּךָ֙ וּמִבָּ֣תֶּ֔יךָ וּמֵעֲבָדֶ֖יךָ וּמֵעַמֶּ֑ךָ רַ֥ק בַּיְאֹ֖ר תִּשָּׁאַֽרְנָה׃
וַיַּ֥עַשׂ ה״ כִּדְבַ֣ר מֹשֶׁ֑ה וַיָּמֻ֙תוּ֙ הַֽצְפַרְדְּעִ֔ים מִן־הַבָּתִּ֥ים מִן־הַחֲצֵרֹ֖ת וּמִן־הַשָּׂדֹֽת׃
וַיִּצְבְּר֥וּ אֹתָ֖ם חֳמָרִ֣ם חֳמָרִ֑ם וַתִּבְאַ֖שׁ הָאָֽרֶץ׃
Moshe had proposed to Pharaoh that the frogs would be removed from you, your servants, your houses, your people...and only remain in the river.
But what happened was the dead carcasses remained all over Egypt and rotted and stank. This was why Pharaoh took counsel. It seemed to him like the deal was not kept. But the Torah emphasizes that HaShem did exactly as Moshe had asked. This has an implication about what Moshe prayed for. That what Moshe presented to Pharaoh and what Moshe prayed for were not exactly the same. And this created doubt for Pharaoh.
What this is indicating is the specific name which Moshe was using in his prayers. This same name is what caused the staff of Moshe to change into a serpent and then back again. This theme is repeated throughout the entire story of the Exodus. Like for example, in the plague of darkness which was upon Egypt, for the Jews there was light. And this is the name of G-d that Pharaoh didn't know.
A similar thing happened in the account of Shemot 9:28-35. Pharaoh asked for the thunder and hail to stop. But what he received as a consequence of Moshe's prayer was that in addition to the thunder and hail stopping, the rain also stopped falling on Mitzrayim.
As a consequence of not receiving what he had bargained for, he once again became pensive and after reflecting became irascible and vehement that he would not send the Jewish people as he had said.
