Good question and I just ordered a new under table router to replace my
Bosch 1611, 15 amp plunge router. My 1611 is similar to the current 1619, a
big router. My 1611 is 16 years old and has no modern features other that
it is electric.
Anyway, what you want is a router that you can fine tune the height
adjustment with some sort of adjustment knob after making your quick coarse
height adjustment. Note my mentioning "after" making your Quick coarse
adjustment. You do not want one of those after market threaded tubes with a
knob on it. Those take forever to run up and down from one end of
adjustment to the other end. Something to look at for example is the
adjustment setup on the newest Milwaukee routers the Bosch 1613 or 1619 or
the Triton router. All of these routers and I am sure a slew of others will
work nicely under the table. Some are fixed base, some are plunge base.
Your main concern is to be able to make coarse height adjustments quickly
and then be able to fine tune that adjustment. I also have a Bosch 1617evs
router that would work great in my router table but it is kinda small for
some of the bits that I spin on my router table. If you are turning common
horizontal raised panel bits I strongly suggest one of the 15 amp "aka" 3+
hp routers with VARIABLE speed. You need to slow those big bits down to the
10,000 rpm range.

So today I went down to buy my 6th router and had narrowed the choices down
to the fixed base 5625-20 Milwaukee or the plunge base Triton. I had
considered the Bosch 1619 evs but because of the location of the fine tune
adjustment knob it would be behind the router when turned up side down or on
the left side of the router. Neither of those locations appealed to me.
The Milwaukee is a beast and well made. Simple. Variable speed, fast
coarse adjustment, good fine tune height adjustment knob and the ability to
make fine tune height adjustments "above" the table. It is also a 2 wrench
router which I GREATLY prefer over the single wrench models. BUT, I found
that when the Milwaukee is upside down and you push the coarse adjustment
knob to make an adjustment, you better have a both hands on the router to
keep it from falling out of the fixed base. While the above table height
adjustment is a cool idea I can see the hole in the table top filling with
saw dust and clogging the mechanism even though there is a plastic cover
that opens when you push an adjustment tool through it.

So now I turn to the plunge base Triton, Seems to be designed from the
ground up to be hung under the table or used free hand. While I have read
many mixed reviews about 3 or 4 specific problems, these apparently have all
been addressed and the new production Tritons have been fitted with the
fixes. I ordered the Triton from the Woodcraft store. They were sold out.
The big pluses I see here is that bits can be changed from above the table
top. While this is a 1 wrench router like my current Bosch 1611 the router
will remain in the base during a bit change. While 1 wrench routers sound
simpler to use, you still have to use your other hand to hold the router and
or the router shaft lock. So what have you gained by only using 1 wrench?
With 2 wrench routers, you can loosen or tighten the collet with 1 hand. 2
hands are still needed to make coarse adjustments to the collet. The Triton
having a self locking shaft and not requiring removal of the router from the
table may be somewhere in between the 2 wrench and 1 wrench style as far as
ease is concerned. I suspect one hand will be on the wrench and the other
holding the router cabinet to keep it from moving when loosening or
tightening the collet. The Triton will not fall out of its base when making
coarse adjustments under the table.

Other nice touches that the Triton has over other plunge based routers are,
the adjustable depth turret is second to none and simple to use requiring no
screw driver or wrench for multiple height adjustments. This may or may not
be useful in a router table. The Triton comes with an edge guide for free
hand use.

Routers are generally plunge base or "non-plunge" fixed base.