What is the difference between Support and Lane Support?
Roles in Dota2 are very versatile and fluid, so people come up with a lot of their own definitions and meanings. This is nothing more than an artificial distinction.
Judging from the heroes that the wiki has put into each category, the consensus seems to be that "Lane Support" heroes are those whose primary focus is often to help a carry to farm safely through harass, healing, and lane control. The heroes listed under plain "Support" are more often able to focus on other roles than babysitting or securing a lane, such as jungling, ganking, or controlling creep equilibrium.
I heavily disagree with the heroes listed under the categories, and even the distinction between the two categories to begin with. I'd encourage you to focus on how heroes scale with items and with levels to determine their effectiveness in various roles, rather than what someone has categorized them as.
That's one of the nice things about the game, though- there's very little consensus beyond the obvious and so you're free to play most heroes in a variety of roles and scenarios.
Lane support is also called a babysitter because the hero will basically babysit the ally carry to help him grow faster and without too much danger.
Being a babysitter means:
- observer wards close to the lane
- harassing the enemy (best done with a ranged support like dazzle)
- Denying creeps to disable some xp from the enemy
- Not last-hitting any creeps and creating space for your carry to move around without any danger
- Using the pulling techniques for keeping your own creeps next to your towers
- Using the hero-attack technique (i do not know how it is really called), where you have to stay behind your wave of creeps and right click an enemy hero, so that the enemy creeps start attacking you, and they will move inwards to your lane, leaving them exposed for the carry to dispose of them more safely.
- Providing occasional tp's/healing salves/sentry wards for heroes like Broodmother/Rikimaru/Phantom lancer/Bounty hunter,etc
The Support is usually assisting the whole team, and he's not focused on babysitting even though he might do something similar on the lane. Though the support role is encouraged to get some kind of farm to provide certain items (Mekansm, pipe, etc) that will be necessary to the team, where the babysitter or lane support, focuses on sacrificing a support's money and early game towards a fast rush to success from the main carry.