So to get Lydian, you raise the 4th note, right?

And to get Locrian, you lower the 5th note?

So C Lydian would be A B C D E F# G
And C Locrian would be A B C D E F Gb

Which is the exactly the same
Have I gone wrong somewhere?

Locrian you made a boo boo

Locrian is 1, b2, b3, 4, b5, b6, b7

So in key of C you get: C Db Eb F Gb Ab Bb
locrian is
1,b2,b3,4,b5,b6,b7
so
C Db Eb F Gb Ab Bb C
rawr
Quote by DeSean
So to get Lydian, you raise the 4th note, right?

And to get Locrian, you lower the 5th note?

So C Lydian would be A B C D E F# G
And C Locrian would be A B C D E F Gb

Which is the exactly the same
Have I gone wrong somewhere?

C Lydian = C D E F# G A B

* you wrote A dorian

C locrian = C Db Eb F Gb Ab Bb
shred is gaudy music
Quote by DeSean
So to get Lydian, you raise the 4th note, right?

And to get Locrian, you lower the 5th note?

So C Lydian would be A B C D E F# G
And C Locrian would be A B C D E F Gb

Which is the exactly the same
Have I gone wrong somewhere?

There seems to be three mistakes you have made here.

The first has been pointed out regarding the Locrian Mode.

The second is has also been pointed out - regarding the starting on A when if you are in the key of C as designated by saying C Lydian you start with C. This is important if you want to get it right.

The third is the claim that A B C D E F# G and A B C D E F Gb (or in the key of C = C D E F# G A B C and C D E F Gb A B C) as exactly the same. Yes they both have the F#/Gb note however the first has a G and the second one has an F instead - two different notes that aren't shared.

They also have different step patterns. If we stick with C as the starting point then the first one is...
C to D Whole tone || D to E Whole tone || E to F# Whole tone || F# to G Half tone || G to A Whole Tone || A to B Whole tone || B to C Half tone.
This is the 3rd mode of the major scale (the C Lydian Mode as you correctly stated) since it shares the same "ladder" or step pattern as the Major scale it just starts at a different point along it.

The second scale or mode you mention has a step pattern
Whole tone - Whole tone - Half tone - Half tone - Whole+Half tone - Whole tone - Half tone.
This is some kind of mode of some kind of scale but I'm not sure what it is called.

Hope this helps.