OK, what you are calling the tie rod I would call the inner TRE. It has the ball and socket joint that screws into the rack and the metal shaft that the outer TRE screws on to.
It is the "long" part.
I was under the impression you had changed the outer only. Sorry!
The inners are more labor intensive...

A tiny amount of play between the rack and the housing is normal. If it had no play at all it would bind up when loaded. The play is not really enough to measure. Just a "click" when moved up and down, maybe .020".
If it's flopping around the bushing inside the housing may be worn. Sometimes these bushings are available (on 88 Fieros the racks are not available but the bushings are, brass instead of original plastic). A worn out Fiero rack will show 1/8" of play.
Some times the rack housing itself is the bushing. (cheap Korean cars).
I do not know if the bushings are available for Miata racks. The OEM parts display do not list them separately.

I'm trying to help, not to confuse the situation more.

 

No worries. At least for the Miata, I never considered your nomenclature. There's a tie rod, and the (outer) tie rod end. I believe my '65 Mustang has tie rods, and inner and outer tie rod ends--three pieces. Just saying...

I have another rack in the shed (but I can't remember why). I'll pull it out after work to compare the play. Again, there's no play on the driver's side when gripping the rack shaft near the housing.

Thanks,

 

There's no play in the steering shaft of my spare rack. I guess I know what I'm doing next.

 

The spare rack has no noticeable play. I'm gonna be pissed if replacing the rack and tie rods doesn't fix the shake, but the way things have been going, I'd be mad but not surprised.

 

Alrighty then... I depowered that spare rack, laid it in, and snugged down the bolts. There's no play in the shaft on either side even at full extension. I'm not going to install new outer tie rod ends yet. I want to know if it was, in fact, the steering rack that was the culprit. She won't be road ready for a few more days.

 

I took her out yesterday with the replacement steering rack and new tie rods. The shake is gone. I'll never know if it was a tie rod or the rack itself. I still can't remember when I got that spare rack. Funny thing, I knew exactly where it was in the shed.

Steering is noticeably smoother, even though there's slightly more resistance. I followed the same Workshop Manual procedure when tightening the adjusting cover as I did for the rack I replaced. It's gooder. I like it.

 

Just to close this out... I spent the last two days at Virginia International Raceway. I have GPS data that says I hit 132 mph. The shake is gone.