The term "transistor" is much too vague, so the question "who invented the transistor" is ill posed.
There are different kinds of transistors, which use different principles for controlling electrical currents, and which have been invented by different people at different times.
Lilienfeld has invented 2 kinds of transistors: MESFETs (metal-semiconductor field-effect transistors) and depletion-mode MOSFETs (metal-insulator-semiconductor field-effect transistors) (US patents 1,745,175 and 1,900,018). Of these 2 kinds of transistors, the second is rarely used today, while the first is used mostly in special applications for discrete power devices.
The parent article is very wrong in claiming that the transistors invented by Lilienfeld are those that are most frequently used today, which are the enhancement-mode MOSFETs, which have a very different structure and for which much less of the semiconductor materials are suitable than for the simpler transistors invented by Lilienfeld.
The enhancement-mode MOSFET has been invented in 1960 by Martin M. Atalla and Dawon Kahng from Bell Laboratories (US patents 3,206,670 and 3,102,230).
Bardeen and Brattain have discovered the point-contact transistor, which had been used only for a few years before becoming completely obsolete, but which was the first kind of transistor offered as a commercial product.
William Shockley has invented 2 kinds of transistors, which have been very important and which remain the best in some special applications, the bipolar junction transistor (BJT) and the junction field-effect transistor (JFET) (US patents 2,569,347 and 2,744,970). William Shockley has also the huge merit of developing a theory of the physics of semiconductor materials which allowed everybody to design transistors and many other kinds of semiconductor devices.
Therefore the 3 Nobel winners have invented together 3 kinds of transistors and they are rightly called inventors of some of the kinds of transistors, i.e. point-contact, BJT and JFET.
Lilienfeld was the first who conceived some possible structures for a triode that uses a semiconductor instead of vacuum or gas. This was a great advance, but this still resulted logically from the prior knowledge that one can make diodes using either vacuum or gas or a semiconductor, and one can make triodes using either vacuum or gas. Thus he attempted to fill the missing combination.
The Bell Laboratories are truly guilty of trying to hide the fact that their post-war research on making a semiconductor triode had indeed started with the purpose of making alternatives to the devices invented by Lilienfeld, of which they were well aware. However, they eventually invented 4 different kinds of transistors, all of which had structures and principles of operation very different from the 2 Lilienfeld transistors, even if all 6 kinds can be considered as variants of semiconductor triodes.
The first 3 kinds of transistors that have been invented, the 2 Lilienfeld transistors and the point-contact transistor of Bardeen and Brattain, can be made using pieces of a homogeneous semiconductor material. This is why they have been discovered first.
The next 3 kinds of transistors, invented by Shockley and by Atalla with Kahng, contain P-N junctions, so they were invented only after William Shockley had developed the theory of the P-N junctions.
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