I’m not exactly sure what your sources are, but I do think I can clear up some misconceptions:

For one, yes, as a term, samurai and bushi are related. Bushi are warriors; more specifically, freeborn warriors, who were skilled in the art of mounted combat. Early medieval bushi often had retainers in the form of two kinds of dependent warriors, of which the latter, usually called rōtō or rōjū in sources, were also mounted, but, unlike their masters, did not administer/possess land of their own. Bushi emerged in the 10th century as a rural elite, by conflating the local managerial post with warfare. This makes sense, since part of the job is to be able to defend the locals (and yourself) from bandits, who might be inclined to steal all the local produce. Their emergence was facilitated by the imperial court’s abandonment of a conscription army, which proved inefficient and too expensive to maintain, and by a beginning proto-privatization of land, mostly through the emerging institution of the shōen.

At the time, the term samurai referred to a lesser noble in service to a high-ranked noble; usually these lesser nobles were of the sixth court-rank and/or served as guardsmen to their masters. This is the first terminological association of samurai with the martial. In ca. the 13th century, the usage of samurai to distinguish “nobleman” from “commoner” (or: man of court-rank vs. man without rank) emerged. However, at the same time, the terms kugyō (man of the first three court ranks), taifu (fourth), tenjōbito (fifth) also circulated. The border between the fifth rank and the sixth was the hard-gap that divided the "true" aristocracy from merely being "of rank" (a.k.a., not a commoner): only these had the right to seek an audience with the emperor. The fourth rank was priviliged to appointments to provincial governorships, and the third was the other hard divide: only these were qualified to become members of the Kyoto court's council of state.

Some nobles in the capital, by virtue of lacking a career in the bureaucracy, did choose to specialize in studying warfare instead, and became gathering points (and mediators) for the local warrior elites in rural society. It is not clear who the first were that chose this path, but we assume that this occurred as early as the mid-9th century. By the 12th century, the most notable of these were the lineages of the Taira (the Ise Taira) and Minamoto (the Kawachi Minamoto of the Seiwa Genji lineage: note that there are many more Taira and Minamoto out there who are not warriors whatsoever) which would end up as rivals in the 12th century. Another famous warrior lineage was a minor branch of the Fujiwara which built a powerbase in Northeastern Japan (and whose main lineage was wiped out by Minamoto no Yoritomo in 1189). All three names spread like wildfire through the local elites through intermarriage.

Either way, these urban warrior elites—the Taira of Kiyomori and the Minamoto of Yoshitomo—were the first "samurai" who had an unprecedented career at court due to their involvement in the first military conflicts centered on the matter of imperial succession, making them the first warriors to ascend to the fourth court rank and provincial governorship appointments. This means that by virtue of heritage and social status, they had never been in a position to compete with the Fujiwara, as well as the numerous more fortunate Taira and Minamoto who had managed to secure themselves a career in the upper echelons of the imperial administration. However, the Taira under Kiyomori would subsequently manage to be elevated to a status of parity and enter the ranks of the kugyō, as a result of their pivotal role during the Heiji no ran in 1160, which elevated Kiyomori to arguably being the most powerful man in Japan (and the one who had the military backing to…convince others to accede to his supremacy). Still, the only warriors until Ashikaga rule who ascended to the status of kugyō after the destruction of Kiymori’s lineage was the shogun himself: the Hōjō and other central Kamakura elites were all placed on the fourth and fifth court rank within the imperial status hierarchy, well upholding the traditionally established status differences (although those were certainly no longer representative of actual political power and influence).

That being said, when Yoritomo wiped out the Taira (and the Northern Fujiwara) in the 1180s, he remained as the sole, unmatched warrior leader in Japan, although there were more than enough warriors around who were in service to various court nobles as managers of their landed properties and so on, and not to Yoritomo’s shogunate. This, however, did neither replace the court's supreme authority, nor destroy the extant system of administration. The shogunate, as a government, remained inclined to keep out of affairs that did not involve one of its vassals (gokenin), a state of affairs that only really changed in the aftermath of the Mongol Invasions which prompted securing the authority to mobilize all warriors by the shogunate.

The offices of shugo and jitō that were introduced by this Eastern government, however, were inserted into the extant system of land administration—which, at this time, already was fragmented due to the parceling of land into countless estates which were under direct control of monasteries, shrines, and nobles. The interesting thing here is that the latter, the jitō, was a transformation of an extant position in local administration into the functional same “job” except that it was now guaranteed by the shogunate in Kamakura: there was no change on a local level except for the fact that the “employer” (so, the noble etc. who “owned” the land in question) became unable to dismiss or replace his local “manager” without appealing to the shogunate to strip the jitō of his privileges (for which he needed to provide a valid reason). This made everything just more complicated, with lawsuits concerning tax division of land stretching of decades, and reemerging years after they had been settled. Suffice it to say, over a process of more than a century, the conflict between central elites and the periphery ultimately alienated control of most land from the central court nobility (the owners) in favor of the local elites (the managers).

Nevertheless, for at least the course of the Kamakura period, the imperial court very much retained its positions of privilege; although they had to increasingly deal with local warriors simply not doing their job “properly” which led to temporary solutions reducing their total income. And even after the disintegration of this system, which guaranteed the central elite’s economic power base, in the 14th and 15th century, vestiges of it remained—especially in the close vicinity of Kyoto, the seat of imperial power, where, by virtue of proximity, direct control was more feasible to accomplish. During the Sengoku, even the Edo period, we still may find some village nearby Kyoto under control of the Fujiwara regents. Which, of course, is insignificant if we consider that, for example, the Konoe, one of the Fujiwara lineages serving as regents, in the Kamakura period governed estates such as Shimazu estate, which was so vast that it crossed multiple provincial borders (incidentally, the famous Shimazu clan took their name from this estate, having served as stewards in service to the Konoe during the Kamakura period: ties they upheld for centuries to come).

The big watershed in political influence for the court nobility, though, was the 14th century, which saw several decades of “civil war,” referenced to as the Period of the Southern and Northern Courts, until the reunification of the imperial line through mediation by the Ashikaga in 1392. Court nobles, by then, had greatly suffered in control over their economic assets, mostly due to continuing social change and the emergence of new local elites and forms of organization in rural areas; out of necessity many nobles took to being sponsored by the central warrior elites of the Ashikaga government by serving as their teachers in various cultural practices, in which they were valued as experts due to their status as elites of the cultural center: calligraphy, kemari, poetry, literature, music, and so on. Court nobles in the fourteenth century onward virtually re-defined their identities through specialization in such a discipline as a "family skill," in the same way as a commoner would specialize in some form of artisan skill.

(Suffice it to say, about a century later most of these urban warrior elites would also be ousted by their local representants...)

Unfortunately, since I primarily study warrior society, that’s all I can say, but I hope it cleared up some things, namely that there wasn't exactly a replacement of court with warrior elites in the 12th century, and that business did continue as usual for quite some time, although the relationship between the proprietor (court noble) and the managerial (local warrior) classes became more and more conflict-laden.