You may have inadvertently found the cause so a new starter may very well solve your problem. Do of course check everything else while your at it, battery terminals, earth points for corrosion etc all connections between the starter and working your way backwards. Just ask yourself, is it an electrical issue or a mechanical issue?
If you got the engine started and it is running (from your experience) alright then you appear to be leading yourself in the right direction.
Here is an exert from the Haynes manual in regards to the Starter Motor if you decide to give it a go:
*Disconnect the battery negative cable (radio code will need re-entering when you re-attach the battery
*So that access to the motor can be gained both from above and below, firmly apply the handbrake then jack up the front of the vehicle and support it on axel stands. On 1.1, 1.4 and 1.6 models, to improve access to the motor, remove the air cleaner housing.
*Slacken and remove the two retaining nuts and disconnect the wiring from the starter motor solenoid, Recover the washers under the nuts.
*Undo the three mounting bolts, supporting the motor as the bolts are withdrawn. Recover the washers from under the bolt heads and note the locations of any wiring or hose brackets secured by the bolts.
*Manoeurve the starter motor out from underneath the engine and recover the locating dowels from the motor/transmission (as applicable)
This exert was taken from the Haynes manual, and in no way is it my work
Edit: You can also find an informative video on youtube by searching for "how to test your car's starter and solenoid" and choosing the first video you see (4:41 in length). Not sure if I can do links just yet due to my post count.
If you got the engine started and it is running (from your experience) alright then you appear to be leading yourself in the right direction.
Here is an exert from the Haynes manual in regards to the Starter Motor if you decide to give it a go:
*Disconnect the battery negative cable (radio code will need re-entering when you re-attach the battery
*So that access to the motor can be gained both from above and below, firmly apply the handbrake then jack up the front of the vehicle and support it on axel stands. On 1.1, 1.4 and 1.6 models, to improve access to the motor, remove the air cleaner housing.
*Slacken and remove the two retaining nuts and disconnect the wiring from the starter motor solenoid, Recover the washers under the nuts.
*Undo the three mounting bolts, supporting the motor as the bolts are withdrawn. Recover the washers from under the bolt heads and note the locations of any wiring or hose brackets secured by the bolts.
*Manoeurve the starter motor out from underneath the engine and recover the locating dowels from the motor/transmission (as applicable)
This exert was taken from the Haynes manual, and in no way is it my work
Edit: You can also find an informative video on youtube by searching for "how to test your car's starter and solenoid" and choosing the first video you see (4:41 in length). Not sure if I can do links just yet due to my post count.