A tube cooled reactor is a reactor with a cooling medium flowing through the tubes.
The cooling medium through the tubes can be gas or liquid or boiling water (steam generation). It can also be the reactor feed flowing countercurrent through the tubes in the reactor bed.
The Linde isothermal reactor normally uses boiling water (BFW in, steam out) as cooling medium for exothermic reactions. In theory however the Linde design could also be used with a heating medium through the tubes for endothermic reactions.
Purpose of cooling coils (or heating coils) in a reactor is to minimise exothermic temperature rise (or endothermic temperature drop) through the reactor catalyst bed so as to improve conversion.