I have a feeling this will turn into one of my very long and boreing posts so if you want the answer just skip to the end ..
If you have an earth lead fault or indeed an engine earth lead fault when you try and start it , the huge consumer of power (the starter motor ) will in effect take all the power .. your dash lights would go off , interior lights go off .. basically the starter trys to take all the power through a high resistance lead ... this will lower or cut out the voltage to everything else .. kinda like a flat battery .
Engine earth lead fault and things start to smoke as the car uses any earth it can find .
Its not enough to turn it but it will take away the voltage from everything else on start position only ..
You have said you have dash lights but no crank .. so i would tend to rule that out .
CPS fault .. car will spin over but not fire up .. so thats out .
Ignition lock .. it causes various electrical faults but ive never seen one cause a non start .. so thats out .
The starter needs a 12 volts supply to the solinoid .. this then basically shorts x2 pins inside to send power to the starter .. if a poor connection or a poor solinoid connection (knackered starter basically ) then it will not spin over .
So you have replaced the starter .. unless it was an absoulute fortune then its a refurb unit ..
A refurb is a 15 year old plus starter than someone has taken appart .. replaced what they think is broken with it , bench tested , painted it silver then sold as a refurb ..
No guarentee they have actually fixed the fault with it though .. so a refurb is not always a fix .. it can still be faulty .. in your case .. i think not .
You have a main power lead from the battery to the starter .. its not a one piece lead , there are connection points which can get corroded .. on the gearbox .. think 997 starter lead fault .. its possible .. but i think not .
Immobiliser .. there will be fault codes , it is possible but there needs to be a reason .. if a cab then water ingress .. so check under the passenger seat .. is it wet there .. any water bottles stuck there that might leak ?
From memory the car will spin but not fire up with an immobiliser fault .. but thats from memory so i may be wrong .
Sooo .. what are we left with . the car intermitantly doesnt spin over .. the dash lights stay on .. we will assume the 12 v supply is ok to the solinoid but that is a guess .. i cant see the car after all ..
How about the starter relay ?
Its worth chucking a new one in if only to prove its not this ..
Relay board 2 is behind the rear seats , remove the carpet panel , it pulls upwards in the center .. bend it basically . .. you pull it out then undo the 10 mm nuts and lift up the ally panel.
Relay is position 7 and called start lock on the diagram .
Just my guess anyways .
EDIT ..
Ill just add that any system that has a relay then you use that as the diagnostic point .. 4 pin relay so 30 - 87 will be the switched feed .. if you brigde these wires it will activate the item to be tested .. 85 - 86 are the switched inputs to activate the realy .
30 - 87 .. one is 12 volts and the other the item that needs the power .
85- 86 ..
One will be 12 volts the other an earth when activated .. in this case ignition key in start position . .
If you bridge a starter relay and the car is in gear you WILL be heading off down the road with you in the back of the car .. dont do it .. be safe
Second EDIT ..
Starter relay .