Did you move the patches into the main Links folder before running them? Im not sure if the LSPN ones require that, but I believe the 1.07 and Nvidia ones do. However, I too have once seen that message within the game that tells you to update to 1.07, when I was 100% sure 1.07 was installed and its features worked. On the other hand, I didnt try LSPN under that configuration, so as Echoes points out, there is still always a possibility that - in terms of the login crash - there might still be some conflict there.
Do you remember if you properly installed the game on your current computer and the game's current file location, as opposed to simply copy-pasting the whole Links folder from a previous configuration (whether from a different computer or not)?
The following is pasted from Links Corner's FAQ page (which hasnt been updated for a very long time, I believe), I dont know whether it speaks about a different type of Online play than this one, but it might be worth looking into:
Q. I'm trying to play online but I can't connect. What's wrong?
A. There are two common problems that affect online play.
Firstly you must ensure that any Firewall software you have running is allowing Links to listen for connections. Most firewalls ask if Links should be allowed to access the internet when you try and play online but sometimes this has to be set up manually.
If you connect to the Internet via a router you may have to 'open' some ports. The method varies from router to router and full details are outside the scope of this FAQ. However, the ports required for Links 2003 are as follows:-
2300 to 2400
6073
47624