Can a 10 speed rear derailleur be modified for 9 speed and then modified again to a 10 speed deraileur? I have Shimano RD-5700 105 Rear Derailleur (10-Speed, GS Medium Cage, Black).   I bought it to use on my 9 speed bike, but now what to add new shifters (3x10).  My bike shop wants to replace the cassette and chain, which I have done.   But Rear Derailleur, he wants me to replace as well.  I disagree, but I need to show him in writing.   Please post the web link or manual.   (I am an America in Spain trying to demonstrate to my Spanish bike store).
There is no modification to be done. You can use a 10spd Shimano derailleur with 9spd or 10spd components no problem. If they're telling you that they need to "modify" your 10 speed derailleur because you were previously using it with 9 speed components, bid them good day and go find another shop. However I don't think that's what they're concerned with.

Since you're going to be running a 3x10 setup, my guess is that the reason your bike shop wants to replace your derailleur is the cage length of the current derailleur that you have. 3x10 gives you the widest gear ranges, but with wider gear ranges come longer cages to account for larger capacity, that is the derailleur's ability to take up or deal out extra chain between gearing extremes (big-big, small-small, or close to either). Too short caged of a derailleur can cause the chain to lose tension when nearing the small-small combination, or cause the bike to be unable to shift into gear combinations approaching big-big at all. Neither one of those are a good situation and can potentially damage your bike's drivetrain.

If you're running a 3x10 setup, you likely need at least a medium cage derailleur. If you're running a dinner plate on the back (read: cassette with very low gearing, likely 29t or larger) you may actually need a long cage derailleur, or a mountain bike derailleur.