From your natal details, the Moon was at 26 degrees 24 minutes of Gemini - so Astro is correct there. Gemini is your rising sign and the Moon is in your first House.
Why it should be wrong on the other one, is difficult to say - it's not a '12 hour' issue - you entering 2:15 and the site taking it as 2:15 AM, as that would push it further back in Gemini, not advance it into Cancer.
I've just checked the 0800-horoscope site and I get the same result as you did. They seem to have real issues, as it's just possible you entered the data incorrectly (though I really doubt that) but for both of us to enter the data incorrectly and get the same result is just too remote to be a real chance. Their site is wrong. The error is six hours (too late). And that is the case with all planets and the Ascendant and MC
Checking the info that they present, that's clear from the quoted Julian Day and Sidereal Time (ST) so there seems to be an error in their database/program. Now I double checked with my own natal details and they got that right - so it might be something to do with their data on Tehran.
One other fault that I found is that when you enter your birth time and then put in your place of birth, pressing calculate brings up a check box with the place and country - so far so good BUT it resets the 'minutes' in your birth time to zero. Pressing calculate again calculates the chart for your birth hour but not to the minute, and clearly that could be up to 59 minutes out. You have to reset the minutes before pressing calculate for the second time.
So please don't use this site everyone - it's just not reliable.