The mods button does not appear.
latest.log
rei.log
rei-issues.log

There are mods in the folder.

Is Mod Menu one of the mods in your folder?

Isn't the mod menu mod itself a mod? Shouldn't it be located in the folder?

Shouldn't it say mod menu right in this corner?

What is a Minecraft "thematic edition"? Maybe related to the 'thematic' mod? Seems like something is pretty heavily modifying the menus... (Granted, based on the logs there's no evidence at all that modmenu is present in the mods folder.)

Mod Menu is not loaded. Is it in your mods folder?