I am still a bit worried about the second definition, for the case when the square in question is already occupied in such a way that moving there would mean capturing our own piece. Someone might interpret it as "but you cannot move there, so it is not under attack". Of course in that case the piece already on the square will usually be under attack. (But not if there is a type-discriminating attacker.)

I still would think it would be good to mention that attacks can also be pseudo-legal (if explicitly labeled such), with basically the same definition instead that 'legal capture' in that case would be replaced by pseudo-legal capture.

Some more thoughts about pseudo-legal moves: in complex cases it could be murky what you consider the 'path' taken by the piece. For simple sliders or hoppers this is of course obvious. Even for bent hoppers that change direction on or behind their mount the path is intuitively clear. But Mats Winther has designed some 'bifurcator pieces' that would change direction on the square before they collide with an obstacle. The obstacle then doesn't seem to be in the path, (although it is definitely in the path of other potential moves), and the bifurcator doesn't seem to hop. But yet the obstacle affects the move. The situation could be rescued by the interpretation that the path actually does go over the obstacle, turns 90-degrees (say left) there, and immediately after stepping off the obstacle, 45-degree right again. Then it would actually be a hopper ('crypto-hopper'?), with a straight first-leg, but a bent second leg at an angle.

It is even worse for 'deflecting' sliders, which would bend their trajectory 45 degrees when the square perpendicularly next to a square they are passing through is occupied (and on the opposit side empty), in the direction of that occupant. There the 'zone of influence' is not just the path where the slider could be blocked, but also all squares lying next to it. Still only a sub-set of the board, but it starts to be uncomfortably large.

The worst I could come up with was a variant 'Gravitational Chess', where sliders do not move in straight lines, but every time they pass next to a piece experience its gravitational pull and change direction. If they pass between two occupied squares, the gravitational pull cancels, and they keep going as they went. This arbitrary many times.

The Rook here could move g1-g2-g3(2P cancel)-g4-f5(P)-e5(N)-d4(N)-c3-b3(wP)-a4(wP)-a5(bP)-b6(bP)-c7-d8, with in parentheses the piece that deflects it to that square.

Depending on the actual position, a piece in this CV could have any trajectory. It is like a Tenjiku-Shogi area move of arbitrary many King steps that can arbitrarily change direction for every step. One could say all King tours through empty squares ending on an empty square or opponent are pseudo-legal. But only a very small fraction of those would be legal, depending on whether they make the right bend when they pass next to other pieces.