Shimano calls this 'hidden brake and shift cables.'
https://bike.shimano.com/en-EU/product/component/claris-r2000/ST-R2000-L.html
First on Dura-Ace 7900
There is a list of 10-speed groupsets here:
https://en.wikipedia.org/wiki/Shimano#Road_groupsets
Then:
- Dura-Ace 7900
- Ultegra 6700
- 105 5700
- (Tiagra 4600 does NOT have the hidden cables and is based on an older design.)
- Tiagra 4700
- (Not listed there and irrelevant unless you have hydro disc brakes, but ST-RX400)
7900 and 6700 are almost identical internally (both contain ball bearings). 5700 is a bushing design, as is 4700.
4700 is essentially an R9100/R8000/R7000 design; i.e. the brake cable/lever follows 'SLR EV' design, rather than 'New Super SLR'. It also uses the 11-speed RD pull ratio (along with GRX 10 speed), which pulls more cable and is therefore easier to setup and generally better. So you can use any 11-speed RD with this, or RD-4700/RD-RX400 10-speed RDs. But not the older 10-speed road RDs.
7900 revised the brake pull ratio, so your bike will stop poorly if you mix 7900 with brakes BEFORE 7900/6700/5700/4600/3500/2400, but you can mix it with newer design (symmetrical pivots on the arms, rather than one on the arm and one on the top - symmetrical design is called SLR EV), R9000/R8000/R7100/6800/5800/5810 brakes
I would go for the 4700 brifters rather than any older ones.
In terms of FD, assuming you will use band-on, then 4700 has the long-arm design:
the increased leverage here means it should shift better than the 7900/6700/5700, design
(note that 7900/6700/5700 introduced a new pull ratio, which as indicated above did not propagate to 4600, which was older.)
[also note that the long arm created problems with cable routing on certain braze-on frames, so they updated it again for the current R9100/R8000/R7000 series)