That is a good summary but of course there is much more to the story. A more detailed description of development of the Ta152/153 designs can be found in "The Focke-Wulf Ta 152" by Thomas Hitchcock.
The Ta153 was a completely new design proposed by Tank in the summer of 1942 in response to an RLM requirement for a high performance standard escort fighter with enhanced high altitude performance. The design was to embody all of the Fw190A-D developments to that time to create an all-new, heavily armed airframe around the Jumo 213 or DB 603 power plants. Though the Ta153 looked very similar to other FW designs, all of the components were new and were to incorporate new hydraulically controlled undercarriage (the FW190's were electric) and no less than 6 x 20mm cannons around a single 30mm cannon firing through the spinner. The longer fuselage was purpose-built, doing away with the "plug" concept that lengthened the FW190's frame and would have incorporated new lubricant tank and fuel tanks.
Also in the running during that time were the Messerschmitt developments Me209 and 309. By March 1943, Tank sensed that interest in the Ta153 was waning given the RLMs concerns with the fact that entirely new jigging and production facilities would be needed to produce the aircraft whereas the prime competitor, the Me209, was heavily derived from the 109 with up to 65% of it's airframe components identical to those for the 109G. Messerschmitt no doubt used every opportunity to point this out as well.
In response to this dilemma, Tank recognized that his new fighter would need a new pragmatic approach and so began focusing on utilizing a hybrid design based on the Fw190A airframe. This new design was given the name Ta152. Concurrent with these happenings, there continued to be an RLM interest in extreme high altitude capability.
On August 13, 1943 a meeting was held at the RLM to discuss the merits of the Me209 and Ta153 and to "establish a framework whereby one of these two designs would be produced." At that meeting (a translated transcript is provided by Hitchcock) it was noted that comparative performance figures between the two aircraft showed only "minor differences". There followed a lot of discussion about production conversions and the merits of the proposed designs as well as some posturing among some of the key players favouring one design over the other. Ultimately, Milch ordered that the Ta153 was to be dropped and that work on the Me209 was to proceed immediately.