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In Google, a user clicking delete is treated exactly the same as a written deletion request.
In fact, the law requires that they be the same - "Therefore, an individual can make a request for erasure verbally or in writing. It can also be made to any part of your organisation and does not have to be to a specific person or contact point.". (https://ico.org.uk/)
Only stating you want your data to be deleted leaves the counterpart wiggle room.
A person who may decide to bring suit however should always cite chapter and verse to lay down the line and to indicate that they are very serious about it. Just the fact that you would be citing that article will likely give you a better chance of seeing your request honored. But if your request is refused and you decide to tip off a regulator it won't make all that much of a difference, they will do their own investigation outside of the particular case and may broaden/narrow the scope of that investigation as they see fit.
This has already surprised more than one company by the way, they decided to play fast and loose with a single individual and as a result found their whole infra and processes under review with plenty of things found out of order. Fines were handed out that were higher than what it would have cost to arrange things properly in the first place.