You're right, either A or J was used in '71. It sort of depends on your concerns about originality vs. availability vs what fits most easily into your car (rear mounts are different, but can be adapted as you noted).
J-type tend to be a little less expensive and a little more easily found. A-type can be faster shifting into OD. Most folks seem to feel the J-type solenoid is a better setup, the A-type's is a little primitive, fussier to adjust properly and more exposed to the elements.
In fact, there are two different A-types. The early one has a larger accumulator piston and really snaps into OD quickly. Generally that's the most desirable for racing, plus it's rarer and so it's usually the most expensive. The second A-type was put in the cars mid-TR4 production, and is simply a little softer shifting thanks to a smaller accumulator piston.
There are several versions of the J-type, too, but I don't know the details.
Plus, both A-type and J-type were widely used in other cars, mainly distiguished by the rear mount arrangement. Those may or may not be easily adaptable to your car. On the other hand, it might be cheaper to get one out of a Volvo sedan, for example, if it can be made to work.
With any of the ODs, you might want to budget in the cost of a rebuild. There have been improvements in components that will make either type OD more reliable and durable.
For example, the A-type originally used a mechanism that didn't protect against reversing with the OD engaged, which can damage the OD and/or the gearbox (properly set up inhibitors will prevent this from happening, unless the OD sticks "on"). More modern parts available for the A-type eliminate most concerns. The J-type already uses the more modern design. Also, clutch compounds have vastly improved and there are some other possible internal upgrades.
Most A-type are 22% reduction, but can be modified to 25 or 28%. Most J-type are 25%, a few are 28%, but either can be modified to 22% if that's preferred.
The first two digits of the part number stamped on the Laycock plate attached to the OD tell what reduction it was set up to provide when manufactured. "22xxxx" equals 22%, etc. However, any old OD might have been modified to another reduction during its lifetime.
Particularly the stronger shifting early version A-type and any of the ODs on a modified, higher output motor, might be too strong for aggressive use of 2nd gear OD. One key area of concern is the diff, particularly the IRS setup due to the mountings. In fact, many TR6 only came with 3rd/4th gear OD. It's relatively easy to install a second inhibitor switch to enable 2nd gear OD on any gearbox not fitted with it. Likewise, it's easy to remove 2nd gear OD inhibitor, to prevent using it if it's a concern.
I'm installing an A-type in my TR4 and really looking forward to using it, especially with a 4.1:1 diff and 60 series tires.
Cheers!
Alan