Your engine is 9 cubic inch displacement (148 cc), second design generation, horizontal crankshaft, with a Pulsa-Jet carburetor. It has plain main bearings and a rewind starter. It was made on the 23rd of November, 1998. Here is the operator's manual:
http://bsintek.basco.com/BriggsDocumentDisplay/msvtJR-K_ny7tH217FzoPu.pdf
Here is the illustrated parts list:
http://bsintek.basco.com/BriggsDocumentDisplay/msvtJR-K_ny7tH217FzoPu.pdf

Please do not continue to start your engine by putting petrol in through the spark plug hole - this was a diagnostic test, not a solution. Putting liquid petrol in the cylinder washes the oil off the cylinder wall and is likely to damage the engine if repeated.

Since your engine seems to continue to run once started, and will restart when warm without priming, there are two likely causes of your problem, both carburetor-related. One possibility relates to your carburetor which is a Pulsa-Jet type. The Pulsa-Jet has a diaphragm-type fuel pump which pumps fuel continuously into a small chamber in the top of the fuel tank. It has two fuel pick-up tubes dropping vertically from the carburetor into the tank. The long one picks up fuel for the pump, and the pump drops it into the secondary tank. The short tube picks up fuel from the secondary tank, under suction from the carburetor venturi. It is common for the intake filter/screen at the bottom of the long tube to become blocked by debris. Please check that it is clean. Visually check the bottom of the short tube as well, though this is less likely to be contaminated. It is also common for things to go wrong with the fuel pump diaphragm, which is also the gasket between the carburetor and fuel tank. It has a couple of flap valves which are the valves for the fuel pump. You said you replaced the diaphragm. The most likely single cause of the problem you are having, is that you did not get the diaphragm into place correctly, resulting in the fuel pump failing to prime after the mower has been standing. Please read this page and the following ones, of the overhaul manual:
http://recoveryvehicles.tpub.com/TM-5-4240-501-14P/css/TM-5-4240-501-14P_100.htm
(Ignore the references to automatic choke, which apply only to 92000 series engines, yours is a 91000 series with a manual choke). This should help you to ensure that you have the diaphragm properly installed.

If your problem is not due to a fuel pump fault, the other likely cause is that your manual choke is not working properly. The choke is the horizontal-sliding lever immediately below the speed control lever, to the left of the recoil starter handle. Obviously, it must be fully closed for cold-starting, then left fully open at all other times. You can verify that it is working by removing the air cleaner and looking vertically down into the carburetor air intake. It is a butterfly-type choke, visible near the top of the air intake. Move the choke lever from fully left (wide open) to fully right (completely closed) to verify that it is working properly.

Please post some pics of your mower and engine, as well as a report on what you find when you make these checks.