When you are putting more power into the pedals, you are probably crouching or leaning forward a bit too, to grip your bars and use your gluts. The change in posture will cause you to slide your butt back a bit. That's natural. Fore/aft saddle positioning will find a good medium between all your riding postures with comfort and support for you with all of them.

I do believe though that the forward tilt is causing you to use your arms to stay on your saddle; i.e. without your arms holding you there you would slide right off.

That seat post doesn't look like it's beyond specs, unless it's just a really short seatpost. Without the ability to do a full up fit, one easy question should take care of some of that: How much clearance is there between the top tube and your crotch when you stand over the bike with your feet placed about 1 foot apart? For a "hybrid" bike, that number should be about 2-3 inches, I believe. A general rule is 1+" for a road bike or one that will never leave a prepared surface, 2"-3" for a bike that may see some trails and/or very light "off road" duty, and 3+" for a bike that will see serious off road duty. There are also some more technical geometry issues to cover, but since a "hybrid" bike is more or less based off of a MTB type frame, and similar geometry, 2"-3" should get you "in the ballpark".

To measure, stand over your bike, then pull the front wheel up until it makes contact with your crotch. A good solid contact, so not as hard as you can until you've pulled through to the bone, but not at the merest touch of anything either. (you have some bits down there that can give you a "false reading", depending on the temperature, time of day, etc. you know what I mean.) Measure how far the front wheel comes off the ground. A partner is a big help with this.

If that measurement is about right, then there may be some other issues. However, I do think that saddle needs to be level. That forward dumping ramp could be causing a bunch of problems in your arms, shoulders, and in your positioning on the saddle.