I don't agree with the rest of the comments here about playing Lich against their lineup: even if drafts mattered significantly at 2.1k mmr, the reason you guys lost isn't because you played Lich. Looking at purely drafts, it's pretty clear you guys will get out-late gamed. If the Dire team reaches ~40 minutes with decent farm on Storm, PA or Sven, you guys will get crushed. Therefore, for this game, I think it's very important that you crush the enemy team, as opposed to going even into the early/mid game. Therefore your first 20 minutes are gonna dictate your game, since Lich has significantly less impact from there on, both generally and in this game.
First 2 minutes: I noticed that you traded right clicks with the PA/Kotl. This was a poor idea: the Kotl has an extra salve + regen on your pudge is worth much less than on PA (PA has higher armour + evasion); granted she didn't skill it, but you wanna think about what would occur if she had played it properly. You can certainty harass them but doing it with right clicks isn't really good: getting to level 3 and spamming your Nova is more efficient and effective since you have guaranteed mana (kotl is similar but his nuke is much less reliable). Also during this time, you were sacrificing creeps right in front of the PA/Kotl, which gives them free exp. You can run back to the wave before it reaches the tower if you need to: make sure you don't sacrifice in range of enemy heroes. If you sacrifice it properly, the lane effectively becomes you getting exp for 5 creeps (the sacrificed and the 4 enemy) whilst the enemy gets 3.
When you're the only support, it's almost never a good idea to teleport back to lane; that 100 gold is precious, if you had just walked back, you not only save the 100 gold but you would have also sacrificed another creep.
At around 7.30, you teleport down bottom, but you show yourself immediately. Ignoring the fact your Luna is bad, that normally plays out as the Sven and Sniper just walking away. Basically you wasted your 100 gold tping down there.
I'm not really a fan of your skill build 2 1 3 1 at level 7. I'm not really a Lich player but I think you should go 4 0 2 1 at level 7 since generally you should max nukes early. The value point in armour can be good, but you didn't even use it before you got level 8. If you want to take it, make sure you know why.
Your warding at ~13 minute mark is mediocre, placing those 2 river wards have very little impact; they can be useful if the laning phase is still ongoing and you want to protect your farmers from rotations from the river. However laning stages at 13 minutes are rare + the enemy team only really has one ganker, i.e. no roaming support duo. If for example you placed a ward behind the enemy top tier one, your pudge (who seems to be pretty good) can land an easy hook, your tinker could get free missiles and you got yourself a free tower.
After that, Lich's game impact is severely reduced unless you get a hex. Some general things in team fights, you should stay super far back: you're super fast + you have force. Make sure you frost armour heroes constantly; you have the mana to support it. When fights break out, you just wanna stand back and Q any heroes who have overextended blah blah blah.