[RPG] Would Fire Breath go through Wall of Water

As far as the rules are concerned, wall of water won't affect water whip in any way.

Water whip isn't a ranged weapon attack, and it doesn't deal fire damage, so the only part of wall of water that might make a difference here is that it is "difficult terrain". So what's difficult terrain? It's this:

Every foot of movement in difficult terrain costs 1 extra foot.

However, this is taken from the section about the movement of a creature using its speed on its turn, so it doesn't apply to any and all movement.

So as far as the rules are concerned, water whip will be completely unaffected by wall of water. It's up to you if you want to change this; as always, you can make whatever houserules you deem appropriate.

You may want to use an alternative method from XGtE.

Your image is correct, and you cite all relevant rules. The Grid as a visual aid (DMG 250) is different from what the theatre of mind approach offers and provides different results, but there are other methods to adjudicate areas of effect on a Grid.

Xanathar's Guide to Everything's section about Spellcasting gives insight into this in its subsection about Areas of Effect on a Grid (XGtE 86-88). It offers two alternative options, the Template Method (XGtE 86-87) and the Token Method (XGtE 87-88) which offer different results, so you may choose one. From the Token Method:

Circles. [...] a circular area of effect becomes square in it, whether the area is a sphere, cylinder, or radius. For instance, the 10-foot radius of flame strike, which has a diameter of 20 feet, is expressed as a square that is 20 feet on a side, as shown in diagram 2.3.

You can find diagram 2.3. on page 86.