The engine management light showing when cranking just signifies that there are old error codes registered in the history. I shouldn't read too much into this. Basically it means that you need to clear the old codes. We can walk you through how to do this later.

If the temperature warning light shows along with the engine management light while cranking it signifies low cranking voltage. As yours is not showing the battery is delivering a sufficient voltage under cranking load as far as the ECU is concerned.

How do you know your spark is weak? Did you test the plug? How old is your spark plug and is the gap OK?

Have you actually tested the battery voltage with a multimeter, both ignition off, ignition on and while cranking? This will give you a pretty definitive answer as to what your battery is like before you go jump-starting it.

Also, you didn't answer the question about whether you can hear the fuel pump priming when you turn the ignition on? Does the plug smell of fuel after cranking and attempts at starting? How old is the fuel? If its been laid up 'a while' the fuel may have gone off. Modern part-ethanol (E5/E10) fuels go bad very quickly turning acidic and corrosive in the tank. How long is 'a while'?

Cranking and a spark is all you need, as far as the ignition is concerned, to get something to happen. Even if its all a bit weak, you should at least register the engine attempting to catch, stumble or do something unless you have a fuel problem (which I suspect may be the case).

Do the following in order:

1. Check the battery voltage(s)
2. Check the spark plug and gap
3. Refresh the fuel if you don't know it's age/origin
4. Check to see if the fuel pump is priming
5. Smell the spark plug after a starting attempt

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