The datasheet is INCORRECT! but your understanding is not so far off...
The battery capacity of 7Ah, x its voltage (12V), indicates its stored energy not power. If you need 7A for 1 hour you need to read the fine print, to see if that was the capacity at its "1C" rate (1x its capacity, or drain it in an hour)
Oh wait, there is no fine print on that page, unlike a proper datasheet. So if you need that current you take your chances or look elsewhere.
The Yuasa datasheet shows 7Ah at the 20hour rate, falling to 6.4Ah at the 10 hour rate. So that battery from a reputable manufacturer is rated to last 20 hours at 350ma, falling to 10 hours at only 640 ma. It also tells you that the endpoint they measured to is 1.75V per cell, or 10.5V. But at least it tells you what you're getting.
Lead acid batteries can be tuned for different purposes including high current, so it's possible that the advertised battery will source 7A for 1 hour ... if the vendor is trustworthy. If they are, they may be able to supply a proper datasheet on request. Can't hurt to ask!