chore: update docs and trademarks (#2)

TRADEMARK.md

@@ -2,11 +2,11 @@
 
 Version 1.0 dated Oct. 22, 2024
 
-This trademark policy was prepared to help you understand how to use Brainly trademarks, service marks and logos with the Brainly Gene software.  
+This trademark policy was prepared to help you understand how to use Brainly trademarks, service marks and logos with the Brainly Gene software.
 
 **While some of our software is available under a free and open source software license, that copyright license does not include a license to use our trademark**, **and this Policy is intended to explain how to use our marks consistent with background law and community expectation.**
 
-This policy covers our trademarks and service marks, “Brainly” and “Brainly Gene.”
+This policy covers our trademarks and service marks “Brainly Gene” and “Gene.”
 
 This policy encompasses all trademarks and service marks, whether they are registered or not.
 
@@ -16,10 +16,12 @@ This policy encompasses all trademarks and service marks, whether they are regis
 
 * You also cannot use our logo on your website in a way that suggests that your website is an official website or that we endorse your website. You may not use or register our marks, or variations of them as part of your own trademark, service mark, domain name, company name, trade name, product name or service name.
 
-* You can, though, say you like Brainly and the Brainly Gene software, that you participate in the Brainly Gene community, that you are providing an unmodified version of the Brainly Gene software.
+* You can, though, say you like the Brainly Gene software, that you participate in the Brainly Gene community, that you are providing an unmodified version of the Brainly Gene software.
 
 * Trademark law does not allow your use of names or trademarks that are too similar to ours. You therefore may not use an obvious variation of any of our marks or any phonetic equivalent, foreign language equivalent, takeoff, or abbreviation for a similar or compatible product or service.
 
+* You are not authorized to use the Brainly marks or logos to convey or imply an endorsement by or partnership with Brainly without Brainly’s prior written approval.
+
 ## 2. ACCEPTABLE USES
 
 ### Unmodified code
@@ -28,13 +30,13 @@ When you redistribute an unmodified copy of our software \-- the exact form in w
 
 ### Modified code
 
-If you distribute a modified version of our software, you must **remove** all of our logos from it. To assist you with this, we have removed our logos from our source trees, and include them only in our binaries.You may use our word marks, but not our logos, to truthfully describe the origin of the software that you are providing.  
+If you distribute a modified version of our software, you must **remove** all of our logos from it. To assist you with this, we have removed our logos from our source trees, and include them only in our binaries.You may use our word marks, but not our logos, to truthfully describe the origin of the software that you are providing.
 
 For example, if the code you are distributing is a modification of our software, you may say, “*This software is derived from the source code for Brainly Gene software*.”
 
 ### Statements about compatibility
 
-You may use the word marks, but not the logos, to truthfully describe the relationship between your software and ours. If you wish to use our logos, please contact us to discuss license terms.
+You may use the word marks and Brainly Gene logo s to truthfully describe the relationship between your software and ours. If you wish to use our other logos, please contact us to discuss license terms.
 
 ### Naming Compatible Products
 
@@ -46,10 +48,6 @@ If you wish to describe your product with reference to the Brainly Gene software
 
 * You use the following legend in marketing materials or product descriptions: “Brainly is a trademark of Brainly sp. z o.o. <https://github.com/brainly/gene”>
 
-### User groups
-
-You are not authorized to use our marks as part of a user group name.  You are not authorized to conduct a conference using our marks.
-
 ### No Domain Names
 
 You must not register any domain that includes our word marks or any variant or combination of them.
@@ -60,8 +58,6 @@ When you have the right to use our mark, here is how to display it.
 
 ### Trademark marking and legends
 
-The first or most prominent mention of a mark on a webpage, document, or documentation should be accompanied by a symbol indicating whether the mark is a registered trademark (“®”) or an unregistered trademark (“™”). If you don’t know which applies, contact us.
-
 Place the following notice at the foot of the page where you have used the mark: “*Brainly is trademark of Brainly sp. z o.o.*”
 
 ### Use of trademarks in text

docs/docs-site/theme.config.tsx

@@ -69,7 +69,7 @@ const config: DocsThemeConfig = {
     <Link href="/">
       <>
         <Image src="/logo.svg" alt="Gene" width={30} height={30} />
-        <span className="ml-2 text-2xl font-barlow font-black">Gene</span>
+        <span className="ml-2 text-2xl font-barlow font-black">Brainly Gene</span>
       </>
     </Link>
   ),