Computer Engineer

You should go for B.Tech/BE in computer science or information technology.

To become a computer engineer, you require PCM at +2 stage. Therefore you are not eligible to seek admission in B.Tech CSE. You may evaluate doing BCA. Some private universities allows non-maths students to do BCA.

IT is not a core function of majority of PSU. IT is only a facilitating technology to support core business of PSU. Therefore, in every PSU IT professional have limited growth and career opportunities. However, few government organisation such as Railtel, CDAC, NIC, NISG etc which are dedicated for IT, IT engineers have good growth opportunities.

There are plenty of government/PSUs jobs waiting for you. You must prepare yourself for different competitive exams to get them.
You can check the below mentioned links:
Computer engineer career options:
https://www.shiksha.com/career-as-computer-engineer-cc-31
List of government job available for computer science engineer:
https://www.shiksha.com/career-as-computer-engineer-cc-311
Best of luck.

Computer science engineering deals with the designing of software, looking at algorithms and how the two interface with other software. It is part mathematics, part engineering and a little bit of creative arts all mixed together.
Computer engineering is more of a hybrid role that focuses on how computer science and electrical engineering come together. Computer engineering deals with hardware design. The engineer focuses on both the hardware and software aspects of technology. They study how the two interact together.
According to me computer science engineering has more scope.

BS means Bachelor of Science. It is same as BE. The terminology is different from country to country for different courses.

CSE is a good branch if you have interest in it. Don't expect that you will gain every bit of knowledge after doing B.Tech in CSE unless you are studying in IITs or NITs. If don't make it there don't waste time while preparing for JEE Mains for one more year. Just take admission in some metropolitan city where there or more opportunities. While doing B.Tech make sure that you also attend private coaching classes who will teach you industry related courses like android, PHP and python. Also, opt for online courses on "coursera" app and also you can enroll in CDEEP which is an online teaching program initiated by the IITians.

Relating to jobs, please follow the following link:
http://www.indeed.com/m/jobs?q=Computer+Communication+Engineer

you can do M.Tech after your MCA by giving GATE exam.

Appear for national level entrance exams like JEE, SRMJEE, VITEEE , BITS etc and opt for computer Science or information technology as the stream