Otdfbt

Gendry was born in the Crownlands and should therefore be referred to as Gendry Waters. However, he refers to himself as Gendry Rivers.

Did showrunners forget or is Gendry ignorant of the customs?

For Customs, laws and norms, we have to look to the books instead of the show.

A noble bastard is named after the place where he is raised. Jon was born in Dorne, but rather than "Sand", he was called "Snow" since he was raised in the North. Similarly, King Aegon IV had numerous bastards. Three of the "Great Bastards" were:

1. Daemon Waters (From Princess Daena the Defiant - Aegon IV's cousin and daughter to King Aegon III)


2. Brynden Rivers (From Lady Melissa Blackwood - daughter of Lord of Raventree)


3. Aegor Rivers (From Lady Barba Bracken - daughter of Lord of Stonehedge)

Now these three half brothers bear different names. That is because Daemon was born and raised in King's Landing. Whereas both Aegor and Brynden were born in the Red Keep as well but they both left for Riverlands when their mothers fell out of King's favour and went back to their fathers' castles. Therefore they are both named Rivers because they grew up in the Riverlands. Similarly out of the eight Sandsnakes Prince Oberyn sired, Obara was born in Old Town and spent her early years there. She's still called Sand by virtue of being raised in Dorne, rather than Flowers as is custom for bastards from the Reach. Her Half Sister Nymeria was born in Volantis, yet she's also called a Sand (Although we do not know where she grew up).

However Gendry was unacknowledged by Robert so he had no last name at all, like all common born people. He was only a teenage boy when he found out his heritage and found himself spirited away to Riverlands. According to the Books, that's where he spent the years of the war. According to the Show however, he went from Riverlands to Dragonstone and then to King's Landing again. Now technically, Gendry should have no last name at all, since Robert never acknowledged him. But Robert never acknowledged Mya Stone¹ (eldest of Robert's children) either yet she's called Stone. Also compare that with Edric Storm², Robert's only acknowledged bastard by Delena Florent. He was born in Dragonstone (So qualified for the "Waters" name) but was raised in Storm's End, and therefore he was called Storm rather than Waters.

You're correct in your assumption that if Gendry had had to have a bastard name, "Waters" would have been appropriate given his place of birth and where he was raised. It seems more like one of the cases where Show-writers simply lack a clue as to how things happen in the works they are basing their show on. But let's note, things are not supposed to be consistent in the books either. This is what George R.R. Martin says about it:

Bastard names are given only to bastards with at least one parent of
high birth. So the bastard child of two peasants would have no surname
at all.

Thus a bastard name like "Snow" or "Rivers" is simultaneously a stigma
and a mark of distinction. The whole thing with bastard names is
custom, not law.

The highborn parent can bestow the usual name, a new one of his/her
own devising, or none at all. Most legitimate sons of bastards keep
the bastard name, but there are cases where a later generation fiddles
with it to remove the taint. There's one such case that you will meet
in the next book, a minor character descended from a Waters (a bastard
name along the shores of Blackwater Bay) whose great grandfather
changed the name to Longwaters for just that reason.
_{So Spake Martin - SF, TARGARYENS, VALYRIA, SANSA, MARTELLS, AND MORE}

_{1. Mya is a special case which is understandable since she was Robert's firstborn. Robert was never shy in unofficially claiming her which is why everyone in the Vale knows who she is (And that's presumably why she is called Stone despite no formal acknowledgement). She's the only child Robert kept a relationship with even after he broke up with the mother. There's no indication if Robert tried to maintain the relationship once the Rebellion started and he left the Vale for good. But after he became King, he once had a huge row with Cersei because he wanted to bring Mya home to Red Keep. Cersei implicitly threatened to murder her. Robert viciously beat Cersei for that but he never brought Mya home. For her part, it appears Mya knows who her father (How could she not when everyone knew?) is but resents him for abandoning her. She often invents colourful tales about who her father was when she's cheerful. When she's sombre however, she resentfully speaks of a tall, black haired man who threw her up in the air but was always there to catch her until the day that he wasn't, that betrayal is what she says is the reason she no longer puts her trust in men. In the show, there is no Mya Stone.}

_{2. There is no Edric Storm in the show.}