I am facing problem creating a PNG from EPS

The logo in the Header at the top of every page on my sister's website is actually a 800 x 427 px PNG, but was resized in the page builder by the original developer to 275 x 147 px, which is the size it is being displayed at. The unscaled PNG image is here.  (I'm no longer in contact with that developer).

I want to create a smaller (116px high) version of the logo.

If I insert the PNG logo that is in the header into a webpage and then scale it down in the page builder to the dimensions I want, it looks great, as shown on the test page here.

If I create a logo of same height by creating a 427px high PNG from the original EPS (Logo.eps, which I've attached to this post), upload it to here and then scale it in the page builder to have a height of 116px, then it looks better still, as shown on the same test page – which proves there's nothing wrong with the EPS file I’m working form.

But if I try to create a PNG from the EPS with the dimensions I actually want (116px high) and upload that to a webpage – rather than creating a much larger PNG and scaling it down in the page builder – then the resulting PNG looks blurred – as shown on the same test page.

Yet the the smaller PNG looks sharp in Photoshop; and if I insert it into a Word document, or an email, or into almost any other application it still looks sharp – but if I insert it into a webpage it looks blurred. 

Even if I create a PNG from the EPS of the same size (height 147px) that the logo in the Header is displayed at, and upload that, it still looks blurred – as shown on the same test page. And yet as I said, if I create a PNG with a height of 427px and scale it down in the page builder to the height I want, then it looks very sharp.

But I should be able to create a PNG from an EPS with the size I want and use that. I shouldn't have to upload a much larger PNG and scale it down in the page builder.

Any idea why I'm having to insert a huge PNG and scale it down in order for it to look sharp – and why I can't insert a PNG of the correct dimensions? Is there something wrong with the way I'm creating the PNG from the EPS? I've tried dragging the EPS into Photoshop, telling Photoshop what dimensions I want it to have, setting the colour mode to RGB, and then selecting File > Save for Web. I've also tried Saving for Web directly from Illustrator, with even worse results. 

If there's nothing wrong with the way I'm creating the  PNG, then is there a better workaround than the one  I've come up with, of uploading a much larger PNG than is actually required and scaling it down?