Hi!
The only web based engine I have tried so far are Phaser and LibGDX.

Projects made with phaser:

Silly and Trashy multiplayer online game:

https://turupawn.itch.io/mmocheckerstable

Simple and deep  sumo game multiplier offline:

http://turupawn.github.io/HelloPhaser/

Projects made with libGDX:

Shameful flappy bird clone

https://play.google.com/store/apps/details?id=rosalila.flappy.guacamaya2

Phasers Pros (+) and Cons (-)
+Easy to install (it's just a javascript file) 
+Easy to learn (The documentation is super super) 
+Great community!
- Easy to run and share (it can be hosted by a pubic website) 
- Might need some extra work to port to a native movile app (i haven't done this)

LibGDX Pros (+) and Cons (-)

+Scalable language: Java (I don't like it, but at least it has classes and objects, javascript don't)
+Multiplatform by nature
+Supports 3d
+Great community!
- Dificult to install
- Difficult to learn
-If you want to host the HTML app you may need something bigger than a plain apache server, I have tried tomcat and it works fine 

From what you posted I would totally recommend to you: Phaser

I hope this helps

- Turupawn