I just overlook the appereance stuff and skip right into the important part. And that is Trinity Force vs Lich Bane. I will quote my answer to comment I put into my guide. I will make it shorter a bit. Like this are noted not important parts.

Trinity Force = TF

• + 30 AP
• + 30 AD
• + 250 health and 250 mana
• + 12% movement speed bonus
• + Sheen from 150% basic AD (check in Items chapter the table with values, lvl 18 bonus is 176)
• + 30% AS
• + 15% crit chance
• + 25% to slow from AA
• - Costs 600 more gold than LB

Lich Bane = LB

• + 80 AP
• + 30 MR
• + 350 mana
• + 7% movement speed bonus
• + Sheen from TOTAL AP

The reason why I dont like LB too much with Jax is the survivability and utility stuff. If you take wider look at TF it helps you in almost every stat you need: AD - AA + some spells, AP - spells, movement speed - 437 with T2 boots and TF, health and mana, AS - to stack passive, crit chance - probly the less valuable part of TF, slow, Sheen proc. On the other hand LB gives you only AP and Sheen proc to increase your damage, mana is somehow neutral, movements speed bonus is worse than from TF, MR - ok.

It is true that Rylais + LB + Hextech gives 230 AP, I dont use any runes or masteries for AP so this AP value is final. Lets say other three items will be boots, Warmogs and Atmas, that makes total cca 4000 HP and 250 AD. So now you are super equal Hybrid. If you switch LB for TF, you will have a bit less AP (-50) and higher AD (+30 + little bonus from Atma from additional HP) and you will deffinitely have over 4K HPs (like 4300), but you will be FASTER, with HIGHER BASIC ATTACK SPEED and you will SLOW FROM AAs AND SPELLS.

Now what about Sheen proc? 150% from basic AD is still 176 (talking about lvl 18) and your total AP is 180 (with TF). So this damage proc is same (4 point are not difference here), ok? And additional you have those survivability and utility stuff written above.

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These are the reason why TF is better than LB. Just try. Math doesnt lie. :-)

You shouldn't wave off advice man. Never people to not give you any advice, because you never know what you might have wrong. For example Lich Bane doesn't make any sense with so little AP, and it would make more sense to use Trinity Force on hybrid Jax without Rabadon's Deathcap. Would you listen to me or just keep that in? Also you don't have to put EXACTLY what you have in your runes and masteries page. Just put the most useful runes you can possibly get. Hell when I started making guides I had like no runes. You should've put more conventional runes, because Jax doesn't need all that attack speed. 29 in offense is a total waste, because of all the defense or utility you can use, instead of the unneeded points in Havoc or Archmage .

I also want you to look at this guide:
Making A Guide

That's so you can make the guide look nice, and make it look presentable.

Also if you're laning with Jax, Vampiric Scepter will probably handicap you without a good support around. Might wanna get a Doran's item or an item with potions.