At least here, the ramp agent is not allowed to alter a document signed by the Captain on his own.
On occasion, only the fact that deicing is needed is known before the documents are handed over to the ramp agent and details are decided on later. Consider a not untypical winter weather with snow showers drifting in and out of the airfield - it may well happen that the Captain signs up for deicing out of precaution, but during taxi it is obvious that one will be airborne well before the next shower hits. So it is well possible that depending on airframe contamination either no deicing is needed at all or only washing off the present contamination with some diluted type I will do and no second type II treatment is required. Or if the Captain has underestimated the time it would take to get to the runway and an unexpected snow shower catches him during taxi, he will of course require deicing by calling ATC. Both of these situations are no longer involving any ramp agent, therefore anything written on the documents handed over to him is obsolete. It may well be that he is already working another flight when the final call is made.
Documentation requirements are met by noting the actually performed deicing procedure in the aircraft log on the flight crew side and by keeping a separate record on them on the deicing crew side. The ramp agent is usually not directly involved in either.