Front derailleur cable change
That sequence is more or less correct, though different mechanics will likely do it differently.

First off, if you haven't changed derailers (or adjusted the the one you have) then the low and high limits should be about correct.

The front shifter has a spring that pulls the derailer cage in the direction of the small sprocket, so without the cable in place it will move that direction.  Adjust the LOW limit to center the derailer cage over the low sprocket.

In order to adjust the HIGH limit you need to have something (ie, the cable) pull the cage in that direction, so attach the cable.  To get the cable about right, though, put the shifter on the lowest gear setting, screw in any barrel adjuster, and then stretch the cable tight and clamp (but do not cut the cable short yet).  Then use the shifter to move the derailer to the high end and adjust the HIGH limit to center the cage over the large ring.

Once you've got high and low set, and the cable attached, you adjust the barrel adjuster to center the cage over the middle ring when that's selected with the shifter.  Then repeat the whole process to fine-tune it.  And expect to need to tighten the barrel adjuster again in a few days as the cable stretches out.

I'm skeptical that your setup lacks (and is intended to lack) barrel adjusters.  They may have been incorporated in the old shifters and the new ones lack them, in which case you'd need to buy inline adjusters.  Or the adjusters may be there but "hiding" -- on the derailer body, in one of the frame fittings for the cable, etc.